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\section{Introduction and results} \label{sec:Intro} Consider an $n$-variate polynomial of degree at most $d$: \[ p=\sum_{|\alpha| \le d} c_{\alpha} x^{\alpha} \] where $x:=(x_1,\ldots,x_n)$, $\alpha \in \ZZ_{\ge 0}^n$, $|\alpha|:=\sum_i \alpha_i$, $x^{\alpha}:=\prod_i x_i^{\alpha_i}$, and where each coefficient $c_\alpha$ is taken from a ground field $K$. A {\em determinantal representation} of $p$ is an $N \times N$-matrix $M$ of the form \[ M=A_0 + \sum_{i=1}^n x_i A_i, \] where each $A_i \in K^{N \times N},$ with $\det(M)=p$. We call $N$ the {\em size} of the determinantal representation. Clearly, since the entries of $M$ are affine-linear forms in $x_1,\ldots,x_n$, $N$ must be at least the degree of $p$. Determinantal representations of polynomials play a fundamental role in several mathematical areas: from {\em algebraic geometry} it is known that each plane curve ($n=2$) of degree $d$ over an algebraically closed field $K$ admits a determinantal representation of size $d$ \cite{Dickson,Dixon}. Over non-algebraically closed fields, and especially when restricting to symmetric determinantal representations, the situation is much more subtle \cite{IshitshukaIto}. For larger $n$, only certain hypersurfaces have a determinantal representation of size equal to their degree \cite{Beauville,Dickson}. In {\em optimisation}, and notably in the theory of {\em hyperbolic polynomials} \cite{Wagner}, one is particularly interested in the case where $K=\RR$, $A_0$ is symmetric positive definite, and the $A_i$ are symmetric. In this case, the restriction of $p$ to any line through 0 has only real roots. For $n=2$ the converse also holds \cite{HeltonVinnikov,LewisParriloRamana}; for counterexamples to this converse holding for higher $n$, see \cite{Branden}. In {\em complexity theory} a central role is played by Valiant's conjecture that the permanent of an $m \times m$-matrix does not admit a determinantal representation of size polynomial in $m$ \cite{Valiant}. Via the {\em geometric complexity theory} programme \cite{MulmuleySohoniI} this leads to the study of polynomials in the boundary of the orbit of the $N \times N$-determinant under the action of the group $\GL_{N^2}(K)$ permuting matrix entries. Recent developments in this field include the study of this boundary for $N=3$ \cite{HuttenhainLairez} and the exciting negative result in \cite{BurgisserIkenmeyerPanova} that Valiant's conjecture can {\em not} be proved using occurrence obstructions proposed earlier in \cite{MulmuleySohoniII}. Our motivation comes from {\em scientific computing}, where determinantal representations of polynomials have recently been proposed for efficiently solving systems of equations \cite{BorMichiel}. For this application, it is crucial to have determinantal representations not of a {\em single} polynomial $p$, but rather of all $n$-variate polynomials of degree at most $d$. Moreover, the representation should be easily computable from the coefficients of $p$. Specifically, in \cite{BorMichiel} determinantal representations are constructed for the bivariate case ($n=2$) in which the entries of the matrices $A_0,\ldots,A_n$ themselves {\em depend affine-linearly} on the coefficients $c_\alpha$. This is what we call a {\em uniform determinantal representation} of the generic polynomial $p$ of degree $d$ in $n$ variables; see Section~\ref{sec:Problem} for a precise definition. \begin{ex}[The binary quadric] \label{ex:BinaryQuadric} {\rm The identity \[ c_{00} + c_{10}x + c_{01}y + c_{20} x^2 + c_{11} xy + c_{02} y^2= \det \begin{bmatrix} -x & 1 & 0 \\ -y & 0 & 1 \\ c_{00} & c_{10}+c_{20}x+c_{11}y & c_{01}+c_{02}y\\ \end{bmatrix} \] exhibits the matrix on the right as a uniform determinantal representation of the generic bivariate quadric. \hfill $\clubsuit$ } \end{ex} In applications, the matrix $M$ is used as input to algorithms in numerical linear algebra that scale unfavourably with $N$, such as a complexity of $O(N^6)$. Consequently, we are led to consider the following fundamental question. \begin{que} What is the minimal size $N^*(n,d)$ of any uniform determinantal representation of the generic polynomial of degree $d$ in $n$ variables? \end{que} A construction from \cite{BorMichiel} shows that for fixed $n=2$ and $d \to \infty$ we have $N^*(2,d) \le \frac{1}{4} \, d^2 + O(d)$; this construction is reviewed in Section~\ref{sec:First}. We improve the construction from \cite{BorMichiel} by giving a particularly elegant uniform determinantal representations of bivariate polynomials of size $2d+1$ in Example~\ref{ex:2d}, and of size $2d-1$ in Example~\ref{ex:2d1}. In view of the obvious lower bound of $d$ this is clearly sharp up to a constant factor for $d \to \infty$, although we do not know where in the interval $[d,2d-1]$ the true answer lies. We show in Section~\ref{sec:Numerics} how to use these small determinantal representations of bivariate polynomials for solving systems of equations. Before that, we focus on the asymptotic behaviour of $N^*(n,d)$ for fixed $n$ and $d \to \infty$. In this setting, we derive the following result. \begin{thm} \label{thm:Main} For fixed $n \in \ZZ_{\ge 2}$ there exist positive constants $C_1,C_2$ (depending on $n$) such that for each $d \in \ZZ_{\ge 0}$ the smallest size $N^*(n,d)$ of a uniform determinantal representation of the generic polynomial of degree $d$ in $n$ variables satisfies $C_1 d^{n/2} \le N^*(n,d) \le C_2 d^{n/2}$. \end{thm} We will also compare our results with previous constructions, most notably with those by Quarez \cite[Thm.~4.4]{Quarez}, who proves the existence of a symmetric representation of size ${n+\lfloor \frac{d}{2} \rfloor} \choose n$. For fixed $n$ and $d \to \infty$, \cite{Quarez} therefore has the asymptotic rate $\sim d^n$, meaning that the results of this paper represent a clear improvement. For fixed $d$ and $n \to \infty$, \cite{Quarez} leads to the asymptotic behavior $\sim n^{\lfloor d/2 \rfloor}$, which is similar to our bounds; we will discuss more details in Section~\ref{sec:Outlook}. In Section~\ref{sec:Problem} we formalise the notion of uniform determinantal representations, study their symmetries, and derive some simple properties. In particular, we relate uniform determinantal representations to spaces of singular $N \times N$-matrices. In Section~\ref{sec:Singular} we briefly review some of the existing literature on these singular spaces, and we prove that for $N>4$ there are infinitely many equivalence classes of such objects; this poses an obstruction to a ``brute-force'' approach towards finding lower bounds on $N^*(n,d)$. In Section~\ref{sec:First} we present a first construction, of which however the size is of the order of $d^n$, rather than $d^{n/2}$, for $d \to \infty$. In Section~\ref{sec:Second} we give a more efficient construction and prove Theorem~\ref{thm:Main}. In Section~\ref{sec:Small} we give upper bounds on $N^*(n,d)$ for small $n$ and $d$ and determine $N^*(2,2)$ and $N^*(3,2)$ exactly. We extend representations from scalar to matrix polynomials in Section~\ref{sec:MatPol}. In Section~\ref{sec:Numerics} we give some numerical results that show that for $n=2$ and small $d$ we get a competitive method for computing zeros of polynomials systems. Finally, in Section~\ref{sec:Outlook} we summarise our main conclusions and collect some questions that arise naturally from our work. \section{Problem formulation and symmetries} \label{sec:Problem} In this section we give a formal definition of uniform determinantal representations, and introduce a group that acts on such representations. We also show that a uniform determinantal representation gives rise to a vector space consisting entirely of singular matrices; such spaces are the topic of next section. Let $K$ be a field and fix $d,n \in \ZZ_{\ge 0}$. Let $F_d$ denote the polynomials of degree at most $d$ in the polynomial ring $K[x_1,\ldots,x_n]$. Furthermore, let $p_{n,d}$ be the generic polynomial of that degree, i.e., \begin{equation}\label{eq:polnd} p_{n,d}=\sum_{|\alpha| \le d} c_\alpha x^\alpha, \end{equation} where $x:=(x_1,\ldots,x_n)$, $\alpha \in \ZZ_{\ge 0}^n$, $|\alpha|:=\sum_i \alpha_i$, $x^{\alpha}:=\prod_i x_i^{\alpha_i}$, and where we consider $c_\alpha$ as a variable for each $\alpha$. \begin{de} For $n,d \in \ZZ_{\ge 0}$, a {\em uniform determinantal representation} of $p_{n,d}$ is an $N \times N$-matrix $M$ with entries from $K[(x_1,\ldots,x_n),(c_\alpha)_{|\alpha| \le d}]$, of degree at most 1 in each of these two sets of variables, such that $\det(M)=p_{n,d}$. The number $N$ is called the {\em size} of the determinantal representation. \end{de} To be explicit, we require each entry of $M$ to be a $K$-linear combination of the monomials $1,x_i,c_\alpha,c_\alpha x_i,\ (i=1,\ldots,n, |\alpha| \le d)$. This means that we can decompose $M$ as $M_0+M_1$, where $M_0$ contains all terms in $M$ that do not contain any $c_\alpha$, and where $M_1$ contains all terms in $M$ that do. We will use the notation $M=M_0+M_1$ throughout the paper. When $n$ and $d$ are fixed in the context, we will also speak of a uniform determinantal representation without reference to $p_{n,d}$. Our ultimate aim is to determine the following quantity. \begin{de} For $n,d \in \ZZ_{\ge 0}$, $N^*(n,d) \in \ZZ_{>0}$ is the minimum among all sizes of uniform determinantal representations of $p_{n,d}$. \end{de} This minimal size could potentially depend on the ground field $K$, but the bounds that we will prove do not. Note that in the definition of $N^*(n,d)$ we do not allow terms in $M$ of degree strictly larger than one in the $c_\alpha$. Relaxing this condition to polynomial dependence on the $c_\alpha$ might affect the exact value of $N^*$, but it will not affect our bounds---see Remark~\ref{re:Poly}. Given a uniform determinantal representation $M$ of size $N$, and given matrices $g,h$ in $\SL_N(K)$, the group of determinant-one matrices with entries in $K$, the matrix $gMh^{-1}$ is another uniform determinantal representation of $p_{n,d}$. In this manner, the group $\lieg{SL}_N(K) \times \lieg{SL}_N(K)$ acts on the set of uniform determinantal representations of $p_{n,d}$. Moreover, there exist further symmetries, arising from affine transformations of the $n$-space. Recall that these transformations form the group $\AGL_n(K)=\GL_n(K) \ltimes K^n$ generated by invertible linear transformations and translations. \begin{lm} \label{lm:Affine} The group $\AGL_n(K)$ acts on uniform determinantal representations of $p_{n,d}$. \end{lm} The statement of this lemma is empty without making the action explicit, as we do in the proof. \begin{proof} Let $g \in \AGL_n(K)$ be an affine transformation of $K^n$, and expand \[ p_{n,d}(g^{-1}x,c)=\sum_{|\alpha| \le d} c'_\alpha x^\alpha, \] where the $c'_\alpha$ are linear combinations of the $c_\alpha$. More precisely, the vector $c'$ can be written as $\rho(g)c$, where $\rho$ is the representation of $\AGL_n(K)$ on polynomials of degree at most $d$ regarded as a matrix representation relative to the monomial basis. Now let $M=M(x,c)$ be a uniform determinantal representation of $p_{n,d}$. Then \[ \det(M(g^{-1}x,\rho(g)^{-1}c))=p_{n,d}(g^{-1}x,\rho(g)^{-1}c)=p_{n,d}(x,c), \] i.e., $M(g^{-1}x,\rho(g)^{-1}c)$ is another uniform determinantal representation of $p_{n,d}$. The action of $g$ is given by $M \mapsto M(g^{-1}x,\rho(g)^{-1}c)$. \end{proof} \begin{ex}[The binary quadric revisited] {\rm Take $n=d=2$ and the affine transformation $g(x,y):=(y,x+1)$ with inverse $g^{-1}(x,y)=(y-1,x)$. We have \[ p_{2,2}(g^{-1}(x,y))= (c_{00} - c_{10} + c_{20}) + (c_{01} - c_{11})x + (c_{10} - 2 c_{20})y + c_{02} x^2 + c_{11} xy + c_{20} y^2. \] We find {\footnotesize \[ c'= \begin{bmatrix} c'_{00}\\c'_{10}\\c'_{01}\\c'_{20}\\c'_{11}\\c'_{02} \end{bmatrix} = \begin{bmatrix} 1 & -1 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 &-1 & 0\\ 0 & 1 & 0 &-2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} c_{00}\\c_{10}\\c_{01}\\c_{20}\\c_{11}\\c_{02} \end{bmatrix} =\rho(g)c; \text{ } \rho(g)^{-1}= \begin{bmatrix} 1& 0& 1& 0& 0& 1\\ 0& 0& 1& 0& 0& 2\\ 0& 1& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 1& 0& 0 \end{bmatrix}. \] If we make the substitutions \begin{align*} c_{00} &\mapsto c_{00}+c_{01}+c_{02} & c_{10} &\mapsto c_{01}+2c_{02} & c_{01} &\mapsto c_{10}+c_{11} & x &\mapsto y-1\\ c_{20} &\mapsto c_{02} & c_{11} &\mapsto c_{11} & c_{02} &\mapsto c_{20} & y &\mapsto x \end{align*} in the uniform determinantal representation of Example~\ref{ex:BinaryQuadric}, then we arrive at the matrix \[ \begin{bmatrix} 1 - y& 1& 0\\ -x& 0& 1\\ c_{00} + c_{01} + c_{02}& c_{01} + c_{02} + c_{02}y + c_{11}x& c_{10} + c_{11} + c_{20}x \end{bmatrix} \] whose determinant also equals $p_{2,2}$. \hfill $\clubsuit$ } \end{ex} The action of the affine group will be used in Section~\ref{sec:Small} to determine the exact value of $N^*(n,2)$ for $n=2$ and $3$. We now turn our attention to the component $M_0$ of a uniform determinantal representation $M$. \begin{lm} \label{lm:SingSpace} For any uniform determinantal representation $M=M_0+M_1$ of size $N$, the determinant of $M_0$ is the zero polynomial in $K[x_1,\ldots,x_n]$. Moreover, at every point $\bar{x} \in K^n$, the rank of the specialisation $M_0(\bar{x}) \in K^{N \times N}$ is exactly $N-1$. \end{lm} \begin{proof} The first statement follows from the fact that $\det(M_0)$ is the part of the polynomial $\det(M)$ which is homogeneous of degree zero in the $c_\alpha$; hence zero. By specialising the vector $x$ of variables to a point $\bar{x} \in K^n$, the rank of $M_0$ can only drop, so the rank of $M_0(\bar{x})$ is at most $N-1$. However, if it were at most $N-2$, then after column operations on $M$ by means of determinant-one matrices with entries in $K$ we may assume that $M_0(\bar{x})$ has its last two columns equal to 0. This means that all entries of $M(\bar{x})=M_0(\bar{x})+M_1(\bar{x})$ in these columns are linear in the $c_\alpha$. This in turn implies that any term in the polynomial $\det M(\bar{x})$ is at least quadratic in the $c_\alpha$. But on the other hand $\det M(\bar{x})$ equals $p_{n,d}(\bar{x})$, which is a non-zero {\em linear} polynomial in the $c_\alpha$ (nonzero since not every polynomial of degree at most $d$ vanishes at $\bar{x}$). This contradiction implies that the rank of $M_0(\bar{x})$ is $N-1$. \end{proof} \begin{lm} \label{lm:Dets} If $M=M_0+M_1$ is a uniform determinantal representation of size $N$, then $V \subseteq F_{N-1}$ spanned by the $(N-1) \times (N-1)$-subdeterminants of $M_0$ satisfies $F_1 \cdot V \supseteq F_d$. \end{lm} Here, as in the rest of this paper, by the product of two spaces of polynomials we mean the $K$-linear span of all the products. \begin{proof} Let $D_{ij}$ be the determinant of the submatrix of $M_0$ obtained by deleting the $i$th row and the $j$th column. On the one hand, $\det(M)=p_{n,d}$ is linear in the $c_\alpha$ by assumption, and on the other hand, by expanding $\det(M)$ we see that the part that is homogeneous of degree one in the $c_\alpha$ is \[ \sum_{i,j}(-1)^{i+j} (M_1)_{ij} D_{ij}; \] this therefore equals $p_{n,d}$. Hence any element $q$ of $F_d$ is obtained from the expression above by specialising the variables $c_\alpha$ to the coefficients of $q$. Since each $(M_1)_{ij}$ is then specialised to an element of $F_1$, we find $q \in F_1 \cdot V$. \end{proof} \begin{re} \label{re:Poly} {\rm Note that the proof above still applies if we allow determinantal representations of the form $M_0+M_1+\cdots+M_e$ where $M_r$ is homogeneous of degree $r$ in the $c_\alpha$ and affine-linear in the $x_i$. Since our upper bound in Section~\ref{sec:Second} builds directly on this lemma, the upper bound holds in this more general setting, as well. } \end{re} \section{Spaces of singular matrices} \label{sec:Singular} Let $M=M_0+M_1$ be a uniform determinantal representation of $p_{n,d}$. Writing $M_0=B_0+\sum_{i=1}^n x_i B_i$, Lemma~\ref{lm:SingSpace} implies that the linear span $\langle B_0,\ldots,B_n \rangle_K \subseteq K^{N \times N}$ consists entirely of singular matrices (and indeed that this remains true when extending scalars from $K$ to an extension field). There is an extensive literature on such {\em singular matrix spaces}; see, e.g., \cite{DraismaBLMS,FillmoreLaurieRadjavi} and the references therein. The easiest examples are the following. \begin{de} A subspace $\mathcal{A} \subseteq K^{N \times N}$ is called a {\em compression space} if there exists a subspace $U \subseteq K^N$ with $\dim(\langle u^T A \mid A \in \mathcal{A},u \in U\rangle_K)<\dim U$. We call the space $U$ a {\em witness} for the singularity of $\mathcal{A}$. \end{de} Given any two subspaces $U,V \subseteq K^N$ with $\dim V=-1+\dim U$, the space of all matrices which map $U$ into $V$ (acting on row vectors) is a compression space with witness $U$. It is easy to see that these spaces are inclusion-wise maximal among all singular spaces. If $\mathcal{A}$ is a singular matrix space, then so is $g \mathcal{A} h^{-1}$ for any pair $(g,h) \in \GL_N(K) \times \GL_N(K)$. We call the latter space {\em conjugate} to the former. \begin{ex} \label{ex:SmallSingSpace} {\rm For $N=2$, every singular matrix space is a compression space, hence conjugate to a subspace of one of the two spaces \[ \left\{\begin{bmatrix} * & * \\ 0 & 0 \end{bmatrix} \right\}, \left\{\begin{bmatrix} 0 & * \\ 0 & *\end{bmatrix} \right\}, \] where the $*$s indicate entries that can be filled arbitrarily. A witness for the first space is the span $\langle e_2 \rangle$ of the second standard basis vector, and a witness for the second space is $K^2$. For $N=3$, there are four conjugacy classes of inclusion-maximal singular matrix spaces, represented by the three maximal compression spaces \[ \left\{\begin{bmatrix} * & * & * \\ * & * & * \\ 0 & 0 & 0\end{bmatrix} \right\}, \left\{\begin{bmatrix} * & * & * \\ 0 & 0 & * \\ 0 & 0 & *\end{bmatrix} \right\}, \left\{\begin{bmatrix} 0 & * & * \\ 0 & * & * \\ 0 & * & *\end{bmatrix} \right\}, \] and the space of skew-symmetric $3 \times 3$-matrices \cite{EisenbudHarris}; the latter is not a compression space. For $N=4$, there are still finitely many (namely, $10$) conjugacy classes of inclusion-maximal singular matrix spaces \cite{EisenbudHarris,FillmoreLaurieRadjavi}, but this is not true for $N \ge 5$, as Theorem~\ref{thm:Sing5} below shows. This theorem is presumably folklore; we include a proof since we have not been able to find a literature reference for it. \hfill $\clubsuit$ } \end{ex} \begin{prop} \label{prop:Max} Assume that $K$ is algebraically closed. For any $m$ and $N\in \ZZ_{\ge 0}$ the locus $X_m$ in the Grassmannian $\Gr(m, K^{N \times N})$ of $m$-dimensional subspaces of $K^{N \times N}$ consisting of all {\em singular} subspaces is closed in the Zariski topology. Moreover, the locus $U_m$ in $X_m$ consisting of all {\em inclusion-wise maximal} singular subspaces is open inside $X_m$. \end{prop} \begin{proof} The first statement is standard. For the second statement, consider the incidence variety \[ Z:=\{(\mathcal{A},\mathcal{A'}) \in X_m \times X_{m+1} \mid \mathcal{A} \subseteq \mathcal{A'} \} \subseteq X_m \times X_{m+1}, \] which is a closed subvariety of $X_m \times X_{m+1}$. The projection of $Z$ into $X_m$ is the complement of $U_m$, and it is closed because $X_{m+1}$ is a projective variety. \end{proof} \begin{thm} \label{thm:Sing5} Assume that $K$ is infinite and of characteristic unequal to two. For $N \ge 5$ there are infinitely many conjugacy classes of inclusion-wise maximal singular $N \times N$-matrix spaces. \end{thm} \begin{proof} Take $N \ge 5$. For sufficiently general skew-symmetric matrices $A_1,\ldots,A_N \in K^{N \times N}$ set $A:=(A_1,\ldots,A_N)$ and define the space \[ \cB_A:=\{(A_1x|\cdots|A_Nx) \mid x \in K^N\} \subseteq K^{N \times N}. \] Each matrix in this space is singular, since for $x \neq 0$ we have \[ x^T(A_1x|\cdots|A_Nx)=(x^TA_1x,\ldots,x^TA_Nx)=0. \] In \cite{FillmoreLaurieRadjavi} it is proved that, for a specific choice of the tuple $A$, the space $\cB_A$ is maximal among the singular subspaces of $K^{N \times N}$. By Proposition~\ref{prop:Max}, $\cB_A$ is maximal for sufficiently general $A$, as well (note that we may first extend $K$ to its algebraic closure to apply the proposition). In the notation of that proposition, we have a rational map \[ \phi:S^N \dashrightarrow U_N,\quad A \mapsto \cB_A, \] where $S \subseteq K^{N \times N}$ denotes the subspace of skew-symmetric matrices; the dashed arrow indicates that the map is defined only in an open dense subset of $S^N$. For any nonzero scalar $t$, $\phi(tA)=\phi(A)$. We claim that, in fact, the general fibre of $\phi$ is indeed one-dimensional. As the fibre dimension is semicontinuous, it suffices to verify this at a particular point where $\phi$ is defined. We take $A_i=E_{i,i+1}-E_{i+1,i}$ for $i=1,\ldots,N-1$ and $A_N$ general; here $E_{ij}$ is the matrix with zeros everywhere except for a 1 at position $(i,j)$. Let $B \in S^N$; if $\phi(A)=\phi(B)$, then there exists an invertible matrix $g \in \GL_N(K)$ such that \[ (A_1 g x|\cdots|A_N g x)=(B_1 x |\cdots|B_N x) \] for all $x$, so that $A_i g =B_i$. Using skew-symmetry of $A_i$ and $B_i$, we find that $A_ig=g^T \! A_i$. Substituting our choice of $A_i$ for $i \in \{1,\ldots,N-1\}$ yields $g_{i,j}=g_{j,i}=0$ for all $j$ with $|i-j|>1$, $g_{i,i+1}=-g_{i,i+1}$, so $g_{i,i+1}=0$ since $\cha K \neq 2$, and $g_{i,i}=g_{i+1,i+1}$. Hence $g$ is a scalar multiple of the identity. It follows that the fibre of $\phi$ through $A$ is one-dimensional as claimed. Since $\dim S=\binom{N}{2}$, we have thus constructed an $(N\binom{N}{2}-1)$-dimensional family inside $U_N$. Given any point $\mathcal{A}$ in $U_N$, its orbit under $\GL_N(K) \times \GL_N(K)$ has dimension at most $2 \, (N^2-1)$ (scalars act trivially). Now for $N=5$ we have \[ N \, \binom{N}{2}-1=5\cdot 10 -1=49 \quad \text{ and } \quad 2 \, (N^2-1)=48, \] so that we have found (at least) a one-parameter family of conjugacy classes of singular spaces. For $N>5$ the difference between $N\binom{N}{2}-1$ and $2 \, (N^2-1)$ is even larger. \end{proof} For large $N$ it seems impossible to classify maximal singular matrix spaces. The construction above already gives an infinite number of conjugacy classes, but there are many other sources of examples. For instance, for infinitely many $N$ there exists a maximal singular matrix space in $K^{N \times N}$ of constant dimension $8$, at least if we assume that $K$ has characteristic 0 \cite{DraismaBLMS}. On the other hand, if the singular matrix space $\mathcal{A}$ has dimension at least $N^2-N$, then it is a compression space with either a one-dimensional witness or all of $K^N$ as witness \cite{Dieudonne} (and hence of dimension exactly $N^2-N$). A sharpening of this result is proved in \cite{FillmoreLaurieRadjavi} (see also \cite{deSeguinsPazzis}). It should be noted that in many cases not even the dimension of such singular matrix spaces is known, for fixed values of the size and rank of the matrices. There is a considerable body of work devoted to giving lower and upper bounds for such dimensions, both in the case of bounded and constant rank, but these bounds are rarely sharp, see, among many other references, \cite{Flanders, IlicLandsberg, Sylvester, Westwick} and the more recent works on skew-symmetric matrices of constant rank \cite{BoraleviFaenziMezzetti,ManivelMezzetti}. Hence the fact that $M_0$ represents a singular matrix space of dimension (at most) $n+1$ does not much narrow down our search for good uniform determinantal representations, except in small cases discussed in Section~\ref{sec:Small}. However, for our constructions in Sections~\ref{sec:First} and~\ref{sec:Second} we will only use compression spaces where the witness has dimension 1 or about $\frac{1}{2} N$, respectively; and our lower bounds on $N^*(n,d)$ are independent of the literature on singular matrix spaces. \section{A first construction} \label{sec:First} In this section we restrict our attention to determinantal representations $M=M_0+M_1$ where $M_0$ represents a compression space with a one-dimensional witness (or, dually by transposition, with a full-dimensional witness). Under this assumption we will prove quite tight bounds on the minimal size of a uniform matrix representation. The following fundamental notion will be used throughout below. \begin{de} We say that a subspace $V \subseteq K[x_1,\ldots,x_n]$ is {\em connected to $1$} if it is nonzero and its intersections $V_e:=V \cap F_e$ satisfy $F_1 \cdot V_e \supseteq V_{e+1}$ for each $e \ge 0$. \end{de} Note that this implies that $V_0=\langle 1 \rangle$. We borrow the terminology from the theory of border bases \cite{LLMRT}, where a set $S$ of monomials is called connected to $1$ if $1 \in S$ and each nonconstant monomial in $S$ can be divided by some variable to obtain another monomial in $S$. The linear span of $S$ is then connected to $1$ in our sense. Translating monomials to their exponent vectors, we will call a subset $S$ of $\ZZ_{\ge 0}^n$ {\em connected to $0$} if it contains 0 and for each $\alpha \in S \setminus \{0\}$ there exists an $i$ such that $\alpha-e_i \in S$, where $e_i$ is the $i$-th standard basis vector. Let $V$ be a finite-dimensional subspace of $K[x_1,\ldots,x_n]$ connected to 1. Choose a $K$-basis $f_1,\ldots,f_m$ of $V$ whose total degrees increase weakly. For each $i=2,\ldots,m$ write \[ f_i=\sum_{j<i} \ell_{ij} f_j \] for suitable elements $\ell_{ij} \in F_1$. Let $M_V$ be the $(m-1) \times m$-matrix whose $i$th row equals \[ (-\ell_{i1},-\ell_{i2},\ldots,-\ell_{i,i-1},1,0,\ldots,0). \] Note that $M_V$ depends on the choice of basis, but we suppress this dependence in the notation, since the property of $M_V$ in the next lemma does not depend on the choice of basis. \begin{lm} \label{lm:Block} The $K$-linear subspace of $K[x_1,\ldots,x_n]$ spanned by the $(m-1) \times (m-1)$-subdeterminants of $M_V$ equals $V$. \end{lm} \begin{proof} By construction, $M_V$ has rank $m-1$ over the field $K(x_1,\ldots,x_n)$ and satisfies $M_V \cdot (f_1,\ldots,f_m)^T=0$. By (a version of) Cramer's rule, the kernel of $M_V$ is also spanned by $(D_1,-D_2,\ldots,(-1)^{m-1}D_m)$ where $D_j$ is the determinant of the submatrix of $M_V$ obtained by removing the $j$th column. So these two vectors differ by a factor in $K(x_1,\ldots,x_n)$. Since $D_1=1=f_1$ we find that they are, in fact, equal. Hence $\langle D_1,\ldots,D_m \rangle=V$ as claimed. \end{proof} We can now formulate our first general construction. This generalises a construction from \cite{BorMichiel} to the multivariate case. \begin{prop} \label{prop:Cons1} Let $V$ be an $m$-dimensional space connected to $1$ and suppose that $F_1 \cdot V \supseteq F_d$. Then there exists a uniform determinantal representation of size $m$ for the generic polynomial of degree $d$ in $n$ variables. \end{prop} \begin{proof} Let $D_1,\ldots,D_m \in K[x_1,\ldots,x_n]$ be the $(m-1)\times(m-1)$-subdeterminants of $M_V$. By $F_1 \cdot V=F_d$ and Lemma~\ref{lm:Block} we can find, for each monomial $x^\alpha$ of degree at most $d$, affine-linear forms $l_{\alpha 1},\ldots,l_{\alpha m} \in F_1$ such that $x^{\alpha}=\sum_j (-1)^{j-1} l_{\alpha j} D_j$. Define \[ s:=\sum_{|\alpha| \le d} c_\alpha (l_{\alpha 1},\ldots,l_{\alpha m}), \] a row vector of bi-affine linear forms in the $x_i$ and the $c_{\alpha}$. Then, by Laplace expansion along the first row, we find that the determinant of \[ \begin{bmatrix} s\\M_V \end{bmatrix} \] is the generic polynomial of degree $d$ in $x_1,\ldots,x_n$. \end{proof} \begin{ex}\label{ex:lin1} {\rm For $n=2$ the following picture gives a space $V$, connected to $1$ and spanned by the monomials marked with black vertices, such that $F_1 \cdot V=F_6$: \begin{center} \includegraphics{tree} \end{center} This is the construction of \cite{BorMichiel}, which shows that there exists a uniform determinantal representation of the generic bivariate polynomial of degree $d$ of size $\frac{1}{4} \, d^2 + O(d)$ as $d \to \infty$. \hfill $\clubsuit$ } \end{ex} The bivariate case generalises as follows. \begin{thm} \label{thm:Cons1} For fixed $n$, there exists a determinantal representation $M=M_0+M_1$ of the generic $n$-variate polynomial of degree $d$ of size $\frac{1}{n \cdot n!} \, d^n + O(d^{n-1})$ such that, moreover, the singular matrix space represented by $M_0$ is a compression space with a one-dimensional witness. Moreover, under this latter additional condition on $M_0$, the bound is sharp. \end{thm} \begin{proof} Note that $\dim F_d=\binom{n+d}{n}= \frac{1}{n!} \, d^n + O(d^{n-1})$. Hence by Proposition~\ref{prop:Cons1} it suffices to show the existence of a subspace $V \subseteq F_d$ connected to $1$ and such that $F_1 \cdot V=F_d$, where $\dim V=\frac{1}{n} \, \dim F_d + O(d^{n-1})$. We will, in fact, show that $V$ can be chosen to be spanned by monomials. First, recall that there exists a lattice $\Lambda$ in $\ZZ^{n-1}$ such that $\ZZ^{n-1}$ is the disjoint union of $\Lambda$ and its cosets $e_i+\Lambda$ for $i=1,\ldots,n-1$, namely, the root lattice of type $A_n$ generated by the rows of the $(n-1) \times (n-1)$-Cartan matrix \[ \left[ \begin{array}{rrrrr} 2&-1& & &\\ -1&2&-1& &\\ &\ddots&\ddots&\ddots&\\ & &-1&2&-1\\ & & &-1&2 \end{array} \right], \] where the empty positions represent zeros \cite[Planche 1]{Bourbaki}. In particular, the index of $\Lambda$ in $\ZZ^{n-1}$ equals $n$. For example, if $n=3$, here is the root lattice $\Lambda$ (in black) and its two cosets (in gray and white): \begin{center} \includegraphics[scale=.5]{A2} \end{center} Now let $\Delta_d$ be the simplex in $\RR^n$ with vertices $0,de_1,\ldots,de_n$, for $i=1,\ldots,n$ let $S_i$ be the set of lattice points in $\Delta_d$ that have $i$th coordinate zero, and set $S_0:=(\ZZ \times \Lambda) \cap \Delta_d$. Define \[ S:=S_1 \cup S_2 \cup \ldots \cup S_n \cup S_0, \] a subset of the lattice points in $\Delta_d$. We claim that $S$ is connected to 0. Indeed, for each $i=1,\ldots,n$ the set $S_i$ is connected to $0$, and from each point $\alpha$ in $S_0$ one can walk within $S_0$ to $S_1$ by subtracting $\alpha_1$ times an $e_1$. Next, we claim that for each $\alpha \in \Delta_d \cap \ZZ^n$ there exists a $\beta \in S$ with $\alpha-\beta \in \{0,e_2,\ldots,e_n\}$. Indeed, there is a (unique) $\beta'$ with this property in $\ZZ \times \Lambda$. If this $\beta'$ has nonnegative entries, then set $\beta:=\beta' \in S_0$. Otherwise, $\alpha$ itself has a zero entry, say on the $i$th position, and we set $\beta:=\alpha \in S_i$. Furthermore, for $i=1,\ldots,n$ the set $S_i$ contains $O(d^{n-1})$ vertices, and $S_0$ contains $\frac{1}{n} \cdot \frac{1}{n!} \, d^n + O(d^{n-1})$ vertices. This concludes the construction--note that in the construction of Proposition~\ref{prop:Cons1} the matrix $M_0$ has a zero row, so that it represents a compression space with a one-dimensional witness. For sharpness, assume that $M=M_0+M_1$ is a uniform determinantal representation of size $N$ such that the singular matrix space represented by $M_0$ is a compression space with a one-dimensional witness. After a choice of basis of $K^n$, we may assume that the first row of $M_0$ is identically zero; write $M_0=[0|M_0']^T$ accordingly. Let $u$ be the first row of $M_1$ and write $M_1=[u|M_1']^T$. Then we have \[ p=\sum_{|\alpha| \le d} c_\alpha x^\alpha = \det[u|M_0'+M_1']. \] Let $D_1,\ldots,D_N$ denote the $(N-1) \times (N-1)$ subdeterminants of $M_0'$. By Lemma~\ref{lm:Dets}, the space $V$ spanned by these satisfies $F_1 \cdot V \supseteq F_d$. This already gives a lower bound of $V$ equal to $d^n/((n+1)n!) + O(d^{n-1})$. To improve the $n+1$ in the denominator into an $n$, we observe that by Cramer's rule, the map \[ F_1^N \to K[x_1,\ldots,x_n],\ (\ell_1,\ldots,\ell_N) \mapsto \sum_i (-1)^i \ell_i D_i \] has every column of $M_0'$ in its kernel. These columns are linearly independent over $K$ (indeed over $K(x_1,\ldots,x_n)$; see Lemma~\ref{lm:SingSpace}). We conclude that \begin{equation} \label{eq:Ineq} N \cdot \dim F_1 - (N-1) \ge \dim F_d, \end{equation} so that \[ N \ge ((\dim F_d)-1)/n = d^n/(n \cdot n!) + O(d^{n-1}), \] as desired. \end{proof} In the next section we derive a second general construction of uniform determinantal representations, which we use to prove Theorem~\ref{thm:Main}. \section{A second construction} \label{sec:Second} For a while, we believed that the uniform determinantal representations of Theorem~\ref{thm:Cons1} were optimal. But then we realised that if one relaxes the condition that $M_0$ represent a compression space with one-dimensional witness to the condition that $M_0$ represent {\em some} compression space, smaller-size representations are possible. The basic example is the following. \begin{ex} \label{ex:2d} {\rm Let $p=\sum_{i+j \le 4} c_{ij} x^i y^j$ be the generic polynomial of degree $d=4$ in $n=2$ variables. It has the following uniform determinantal representation: \begin{equation}\label{eq:repjan} p=\det \begin{bmatrix} -x & \ph-1 & & & & & & & \\ & -x & \ph-1 & & & & & & \\ & & -x & \ph-1 & & & & & \\ & & & -x & 1 & & & & \\ c_{00}&c_{10}&c_{20}&c_{30}&c_{40}& -y & & & \\ c_{01}&c_{11}&c_{21}&c_{31}& & \ph-1 & -y & & \\ c_{02}&c_{12}&c_{22}& & & & \ph-1 & -y & \\ c_{03}&c_{13}& & & & & & \ph-1 & -y \\ c_{04}& & & & & & & & \ph-1 \end{bmatrix}, \end{equation} where the empty positions denote zeros. Let $M=M_0+M_1$ be the matrix on the right-hand side. In this case, $M_0$ represents a compression space with witness $U=\langle e_5,\ldots,e_9 \rangle_K$, which is mapped into $\langle e_6,\ldots,e_9 \rangle_K$. To verify the identity above without too many calculations, note that the 5 maximal subdeterminants of the $4 \times 5$-block with $x$'s are, consecutively, $1,-x,x^2,-x^3,x^4$, and similarly for $y$. The matrix obtained from $M$ by deleting the column corresponding to $x^i$ and the row corresponding to $y^j$ has determinant $x^iy^j$. This example extends to a uniform determinantal representation of size $2d+1$ for the generic bivariate polynomial $p$ of degree $d$. We get $p=\det(M)$, where \[ M=(-1)^d\begin{bmatrix}M_x & 0\cr L & M_y^T\end{bmatrix}, \] $M_x$ and $M_y$ are $d\times (d+1)$ matrices with 1 on the first upper diagonal and $-x$ and $-y$ respectively on the main diagonal, while $L$ is a $(d+1)\times (d+1)$ triangular matrix such that $\ell_{ij}=p_{j-1,i-1}$ for $i+j\le d+2$ and 0 otherwise. Note that we will slightly improve on the size $2d+1$ in Example~\ref{ex:2d1}. \hfill $\clubsuit$ } \end{ex} Example~\ref{ex:2d} generalises as follows. \begin{prop} \label{prop:Cons2} Let $V,W \subseteq K[x_1,\ldots,x_n]$ be subspaces connected to $1$ such that $F_1 \cdot V \cdot W \supseteq F_d$. Then there exists a uniform determinantal representation of the generic $n$-variate polynomial of degree $d$ of size $-1 + \dim V + \dim W.$ \end{prop} \begin{proof} Set $m_1:=\dim V$ and $m_2:=\dim W$. Consider the matrix \[ M:= \begin{bmatrix} M_V & 0 \\ L & M_W^T \end{bmatrix}, \] with $M_V$ and $M_W$ the matrices of sizes $(m_1-1) \times m_1$ and $(m_2-1)\times m_2$ from Lemma~\ref{lm:Block}, and where $L=(\ell_{ij})_{ij}$ is an $m_2 \times m_1$-matrix to be determined. Note that the determinant of $M$ is linear in the entries of $L$. Indeed, setting $L=0$ yields the singular matrix $M_0$, so $\det(M)$ contains no terms of degree 0 in the entries of $L$. Furthermore, deleting from $M$ two or more of the first $m_1$ columns from $M_V$, we end up with a matrix that is singular since, when acting on rows, it maps the span of $\langle e_1,\ldots,e_{m_1-1} \rangle$ into a space of dimension at most $m_1-2$, so $\det(M)$ does not contain terms that are of degree $> 1$ in the entries of $L$. Hence the determinant equals $\sum_{ij} \pm \ell_{ij} D_j E_i$ where the $D_j$ are the maximal subdeterminants of $M_V$ and the $E_i$ are the maximal subdeterminants of $M_W$. By Lemma~\ref{lm:Block} we have $V=\langle D_1,\ldots,D_{m_1} \rangle_K$ and $W=\langle E_1,\ldots,E_{m_2} \rangle_K$. Hence the assumption that $F_1 \cdot V \cdot W \supseteq F_d$ ensures that we can choose the $\ell_{ij} \in F_1$ in such a manner that the determinant of $M$ equals the generic polynomial $p$. \end{proof} \begin{ex} \label{ex:2d1} {\rm Example~\ref{ex:2d} can be slightly improved to a representation of size $2d-1$ by taking $V=\langle 1,x,\ldots,x^{d-1} \rangle$ and $W=\langle 1,y,\ldots,y^{d-1} \rangle$; note that, indeed, $F_1 \cdot V \cdot W \supseteq F_d$. A representation of size $2d-1$ for the polynomial $p$ from \eqref{eq:repjan} is \begin{equation}\label{eq:repjan2}p=\det \begin{bmatrix} -x & \ph-1 & & & & & \\ & -x & \ph-1 & & & & \\ & & -x & -1 & & & \\ c_{00}&c_{10}&c_{20}&c_{30}+c_{40} x& -y & \\ c_{01}&c_{11}&c_{21}+c_{31}x & & \ph-1 & -y & \\ c_{02}+ c_{03} y&c_{12}+c_{22}x& & & & \ph-1 & -y \\ c_{13} x+c_{04} y& & & & & & \ph-1 \\ \end{bmatrix}. \end{equation} We do not know whether the factor 2 can be improved. \hfill $\clubsuit$ } \end{ex} \begin{re} {\rm A representation of the form \eqref{eq:repjan2} can also be obtained from the linearisations based on dual basis from \cite{Robol}. There, linearisations of a univariate polynomial are presented that use the basis of the form $\phi_i(x)\psi_j(x)$, where $\phi_i$ and $\psi_j$ are polynomials. If we use the same approach for a bivariate polynomial with the standard basis $\phi_i=x^i$ and $\psi_j=y^j$, we get a representation of the form \eqref{eq:repjan2} up to permutations of rows and columns. } \end{re} We will now prove our main theorem. \begin{proof}[Proof of Theorem~\ref{thm:Main}, lower bound.] For the lower bound on $N^*(n,d)$, let $M=M_0+M_1$ be a size-$N$ uniform determinantal representation of the generic $n$-variate polynomial of degree $d$. Let $D_{ij}$ be the $(N-1) \times (N-1)$-determinant of the submatrix of $M_0$ obtained by deleting the entry at position $(i,j)$. Then the image of the linear map \[ \phi: F_1^{N \times N} \to K[x_1,\ldots,x_n],\ (\ell_{ij})_{i,j} \mapsto \sum_{i,j} (-1)^{i+j} \ell_{ij} D_{ij} \] contains $F_d$ (see Lemma~\ref{lm:Dets}). We claim that $\phi$ has a kernel of dimension at least $N(N-1)$. Indeed, fix any row index $i_0$. If the $D_{i_0,j}$ are all zero, then we obtain an $(n+1)N$-dimensional subspace of $\ker \phi$ by setting all $\ell_{i,j}$ with $i \neq i_0$ equal to zero and choosing the $\ell_{i_0,j} \in F_1$ arbitrarily. If they are not all zero, then the $N-1$ rows with indices $i_1 \neq i_0$ are linearly independent over $K(x_1,\ldots,x_n)$ and hence {\em a fortiori} over $K$. For each such $i_1$ define $\ell^{(i_1)} \in F_1^{N \times N}$ by \[ \ell^{(i_1)}_{ij}:= \begin{cases} (M_0)_{i_1,j} & \text{ if } i=i_0, \text{ and }\\ 0 &\text{otherwise.} \end{cases} \] Then, by Cramer's rule, we have $\phi(\ell^{(i_1)})=0$, and these $N-1$ vectors are linearly independent. Hence, for each $i_0$ we find a subspace of $\ker \phi$ of dimension at least $N-1$, and these subspaces are linearly independent. Thus we find that \[ N^2(n+1)-N(N-1) = N^2 n + N \ge \dim F_d = \frac{d^n}{n!}+O(d^{n-1}), \] from which the existence of $C_1$ follows. \end{proof} \begin{re} {\rm In the proof of the lower bound we have been a bit more careful than strictly needed: without the discussion of the kernel it follows that $N^2 \ge (\dim F_d)/(n+1)$. But one derives a better constant (for $d\to \infty$) by using the kernel. } \end{re} For the upper bound, we first give a simple construction for even $n$. For odd $n$, a trickier analysis is needed (which also applies in the even case); see below. \begin{proof}[Proof of Theorem~\ref{thm:Main}, upper bound for even $n$] Assume that $n=2m$ with $m \in \ZZ_{\ge 0}$. Let $V$ be the space of polynomials in $x_1,\ldots,x_m$ of degree at most $d$, and let $W$ be the space of polynomials in $x_{m+1},\ldots,x_n$ of degree at most $d$. Then $V$ and $W$ are connected to $1$ and we have $F_1 \cdot V \cdot W \supseteq F_d$, so that by Proposition~\ref{prop:Cons2} we have $N^*(n,d) \le -1 + \dim V+\dim W$. Now compute \[ \dim V=\dim W=\binom{m+d}{m}=\frac{d^{n/2}}{(n/2)!}+O(d^{m-1}). \] This implies the existence of $C_2$ for even $n$. \end{proof} We now give a construction that works for all $n>2$, for which we thank Aart Blokhuis. \begin{proof}[Proof of Theorem~\ref{thm:Main}, upper bound for $n>2$] For $i=0,1$ let $B_i \subseteq \ZZ_{\ge 0}$ denote the set of nonnegative integers that can be expressed as $\sum_{j=0}^e b_j 2^{2j+i}$ with $b_j \in \{0,1\}$, i.e., whose binary expansions have ones only at even positions (for $i=0$, counting the least significant bit as zeroeth position) or only at odd positions (for $i=1$). Observe that $B_0+B_1=\ZZ_{\ge 0}$ and that both $B_0$ and $B_1$ contain roughly $\sqrt{d}$ of the first $d$ nonnegative integers for every $d$--they have ``dimension $1/2$''. Now set $A_i:=B_i^n \subseteq \ZZ_{\ge 0}^n$ for $i=0,1$, so that $A_0+A_1=\ZZ_{\ge 0}^n$ and the number of elements of $A_i$ intersected with a large box $[0,d]^n$ is roughly $d^{n/2}$. Unfortunately, $A_0$ and $A_1$ are not connected to $0$. However, we can connect them to 0 as follows. For a lattice point $\alpha \in A_i \setminus \{0\}$ let $l$ be the minimum among the 2-adic valuations of its entries, attained, say, by $\alpha_j$. Then set $\tilde{\alpha}_j:=\alpha_j-2^l \in B_i$. Setting the remaining coordinates of $\tilde{\alpha}$ equal to those of $\alpha$ we have $\tilde{\alpha} \in A_i$ and \[ \|\alpha-\tilde{\alpha}\|_1=2^l. \] We propose to add to $A_i$ the sequence \[ \alpha-e_j,\alpha-2e_j,\ldots,\alpha-(2^l-1)e_j \] to connect $\alpha$ to $\tilde{\alpha}$. We need to verify, however, that in doing this, $A_i$ retains dimension $n/2$. The fraction of $\alpha$ in (a large box intersected with) $A_i$ for which the minimal valuation is at least $l$ equals roughly $(2^{(-l+i)/2})^n$--after all, the condition is that for each $j=1,\ldots,n$, $\alpha_j$ has zeros on the first $(l-i)/2$ positions where it is allowed to have ones. Write $l=i+2m$. Thus by adding the sequences above, the total increase of $A_i$ is by a factor of at most \[ \sum_{m=0}^\infty 2^{2m+i} (2^{-m})^n = \sum_{m=0}^\infty 2^{(2-n)m+i}. \] This is a convergent series as $n>2$, and hence $A_i$ retains dimension $n/2$. We have thus constructed subsets $A_0,A_1 \subseteq \ZZ_{\ge 0}^n$ that satisfy $A_0+A_1=\ZZ_{\ge 0}^n$, are connected to $1$, and that contain roughly a constant times $d^{n/2}$ points in each box $[0,d]^n$. For $i=0,1$ let $V_i$ be the space spanned by the monomials whose exponent vectors lie in $A_i \cap [0,d]^n$. Then $V_1 \cdot V_2 \supseteq F_d$ and $V_1,V_2$ are connected to $1$, so by Proposition~\ref{prop:Cons2} there exists a uniform determinantal representation for $p_{n,d}$ of size $\dim V_1+\dim V_2=O(d^{n/2})$, as desired. \end{proof} \begin{re} {\rm The construction in the proof is by no means tight. For example, one could also replace $\alpha_j$ by $\alpha_j-2^l+2^{l-2}+2^{l-4}+\ldots+2^i$, which yields a shorter sequence to be added; and for large $l$ we have also counted additional, shorter sequences since they also have valuation larger than numbers smaller than $l$. We think that for even $n$ the previous construction, subdividing the variables into two sets of equal size, may lead to a better constant, but we have not verified this. } \end{re} \begin{ex} {\rm Carrying out the construction in the proof for $A_0$ with $n=3$, always choosing for $j$ the smallest index of a coordinate of $\alpha$ with minimal valuation, we arrive at the following fractal-like structure (the circles indicate the points of $A_0$, the black edges show that $A_0$ is connected to $0$): \begin{center} \includegraphics[width=.4\textwidth]{fractal} \end{center} } \end{ex} \section{Small $n$ and $d$} \label{sec:Small} In this section we give several uniform representations of--to our knowledge--the smallest possible size for cases where $n$ and $d$ are small. We start with the two cases where we can compute $N^*(n,d)$ exactly. \begin{prop} $N^*(2,2)=3$. \end{prop} \begin{proof} Taking $V=\langle 1,x,y \rangle$ in Proposition~\ref{prop:Cons1} we see that $N^*(2,2) \le 3$; this is the representation of Example~\ref{ex:BinaryQuadric}. Suppose that a uniform determinantal representation $M=M_0+M_1$ of size $N=2$ exists. Then, by Example~\ref{ex:SmallSingSpace}, after acting with $\SL_2(K) \times \SL_2(K)$ and transposing if necessary, we may assume that the singular space represented by $M_0$ is a compression space with a one-dimensional witness. But then \eqref{eq:Ineq} reads \[ 2 \cdot 3 - 1 = N \cdot \dim F_1-(N-1) \ge \dim F_2 = 6, \] a contradiction. Hence $N^*(2,2)=3$. \end{proof} \begin{prop} $N^*(3,2)=4$. \end{prop} \begin{proof} Taking $V=\langle 1,x,y,z \rangle$ in Proposition~\ref{prop:Cons1} we see that $N^*(3,2)\le 4$. Suppose that a uniform representation of size $N=3$ exists. Up to transposition, there are three possibilities for the singular space $\mathcal{A}$ represented by $M_0$; see Example~\ref{ex:SmallSingSpace} (where the third compression space is conjugate to the transpose of the first): \begin{enumerate} \item Assume that $\mathcal{A}$ is a compression space with a one-dimensional witness, so that after acting with $\SL_3(K) \times \SL_3(K)$ we have \[ M_0=\begin{bmatrix} 0 & 0 & 0 \\ * & * & *\\ * & * & * \end{bmatrix}. \] Let $D_j$ denote the determinant of the minor of $M_0$ obtained by deleting the first row and $j$th column. Then the linear map \[ \Omega:F_1^3 \mapsto K[x,y,z],\ (l_1,l_2,l_3) \mapsto l_1D_1-l_2D_2+l_3D_3 \] has $F_2 \subseteq \im \Omega$. Now inequality~\eqref{eq:Ineq} reads \[ 3 \cdot 4 - 2 = N \cdot \dim F_1-(N-1) \ge \dim F_2 = 10, \] which holds with equality. This means that, in fact, $\im \Omega$ must {\em equal} $F_2$. In particular, $D_1,D_2,D_3$ must all be of degree one (or else $\im \Omega$ would contain cubic polynomials). The image of $\Omega$ depends only on the span $V:=\langle D_1,D_2,D_3 \rangle \subseteq F_1$. If $1 \not \in V$, then there exists an affine transformation in $\AGL_3(K)$ that maps $V$ into a subspace of $\langle x,y,z \rangle$. Then $1 \not \in F_1 \cdot V=\im \Omega$, a contradiction. If $1 \in V$, then after an affine transformation we find $\langle 1 \rangle \subseteq V \subseteq \langle 1,x,y \rangle$. In that case, $z^2 \not \in F_1 \cdot V$, another contradiction. \item Assume that $\mathcal{A}$ is a compression space with a two-dimensional witness, so that after row and column operations we have \[ M_0=\begin{bmatrix} 0&0&q \\ 0&0&r \\ s&t&* \end{bmatrix}, \] where $q,r,s,t \in F_1$. Write $M_1=(m_{ij})_{ij}$. Using that $\det(M)$ is assumed to be linear in the $c_\alpha$s, we find that \[ \det(M)=-m_{11} rt + m_{12} rs + m_{21} qt - m_{22} qs. \] Consequently, setting $V_1:=\langle q,r \rangle$ and $V_2:=\langle s,t \rangle$, we have $F_1 \cdot V_1 \cdot V_2 \supseteq F_2$. If $1 \not \in V_1$, then by acting with a suitable element of $\AGL_3(K)$ we achieve that $V_1 \subseteq \langle x,y,z \rangle$. But then $F_1 \cdot V_1 \cdot V_2 \not \ni 1$. The same applies when $1 \not \in V_2$. On the other hand, if $1 \in V_1 \cap V_2$, then by an element in $\AGL_3(K)$ we achieve that $\langle 1 \rangle \subseteq V_1,V_2 \subseteq \langle 1,x,y\rangle$. In that case, $z^2 \not \in F_1 \cdot V_1 \cdot V_2$. \item Finally, assume that $\mathcal{A}$ is conjugate to a space of skew-symmetric matrices, so that after conjugation \[ M_0= \left[ \begin{array}{rrr} 0 & q & r \\ -q& 0 & s \\ -r&-s & 0 \end{array} \right] \] where $q,r,s \in F_1$. Set $V:=\langle q,r,s \rangle \subseteq F_1$. Then the space spanned by the $2 \times 2$-determinants of $M_0$ is $V \cdot V$, of dimension at most $6$. Moreover, we have $F_1 \cdot V \cdot V \supseteq F_2$. If $1 \not \in V$, then by acting with $\AGL_3(K)$ we achieve that $V \subseteq \langle x,y,z \rangle$, and hence $1 \not \in F_1 \cdot V \cdot V$. If, on the other hand, $1 \in V$, then we achieve that $\langle 1 \rangle \subseteq V \subseteq \langle 1,x,y \rangle$, and $z^2 \not \in F_1 \cdot V \cdot V$. \end{enumerate} In each of these cases we arrive at a contradiction. Consequently, $N^*(3,2)=4$ as claimed. \end{proof} The proofs above use the classification of spaces of small singular matrices in an essential manner, as well as the action of $\AGL_n(K)$ on uniform determinantal representations. We conjecture that $N^*(4,2)=5$, and that this can still be proved in the same manner, using the classification of $4 \times 4$-singular matrix spaces from \cite{FillmoreLaurieRadjavi}. But as Theorem~\ref{thm:Sing5} shows, fundamentally new ideas will be needed to prove lower bounds in larger situations. For some pairs of small $n$ and $d$ we now give the smallest uniform representations that we have been able to find. For the constructions we use Proposition~\ref{prop:Cons2} with subspaces $V,W\subseteq K[x_1,\ldots,x_n]$ spanned by the monomials and connected to $1$. First, we give in Table~\ref{tab:minnd} the minimal sizes known to us of uniform determinantal representations for some small values of $n$ and $d$. \begin{table}[!htbp] \caption{Minimal known sizes of uniform determinantal representations we have been able to construct for $n$-variate polynomials of degree $d$; cf.~Table~\ref{tab:wv}.\label{tab:minnd}} \begin{center} {\footnotesize \begin{tabular}{c|ccccccccc} \hline \rule{0pt}{2.3ex $n$ & $d=2$ & $d=3$ & $d=4$ & $d=5$ & $d=6$ & $d=7$ & $d=8$ & $d=9$ \\ \hline \rule{0pt}{2.3ex 2 & 3 & \ph{1}5 & \ph{1}7 & \ph{1}9 & 11& 13& 15& 17 \\ 3 & 4 & \ph{1}7 & 10 & 14& 18& 22& 27& 34 \\ 4 & 5 & \ph{1}9 &14 & 19& 26& 34& 44 \\ 5 & 6 &11 &18 & 26& \\ 6 & 7 &13 &22 & 33& \\ 7 & 8 &15 &27 & 39& \\ 8 & 9 &17 &32 & \\ \hline \end{tabular} } \end{center} \end{table} The corresponding representations for the entries in Table~\ref{tab:minnd} for $n=2$, which are of size $2d-1$, are given in Example~\ref{ex:2d1}. For $d=2$ we take $V= \langle 1,x_1,\ldots,x_n \rangle$ and $W= \langle 1 \rangle$, therefore $N^*(n,2)\le n+1$, while for $d=3$ we can take $V=W= \langle 1,x_1,\ldots,x_n \rangle$, and hence $N^*(n,3)\le 2n-1$. In Table~\ref{tab:wv} we give sets $V$ and $W$ for the remaining nonzero entries in Table~\ref{tab:minnd}. The subspaces $V$ and $W$ have the form $V=V_0\cup V_1$ and $W=W_0\cup W_1$, where \begin{equation}\label{eq:vw0} \begin{matrix} V_0=\langle1,x_1,\ldots,x_n,\ldots,x_1^e,\ldots,x_n^e\rangle,\\[0.4em] W_0=\langle1,x_1,\ldots,x_n,\ldots,x_1^f,\ldots,x_n^f\rangle \end{matrix} \end{equation} for $e=\lceil (d-1)/2\rceil$ and $f=\lfloor (d-1)/2\rfloor$, which yields $d-1=e+f$. For clarity and brevity, the variables $x,y,z,w,u,v,q,s$ in Table~\ref{tab:wv} stand for $x_1,\ldots,x_8$, respectively. \begin{table}[!htbp] \begin{center} \caption{List of monomials in $V_1$ and $W_1$ that, together with $V_0$ and $W_0$ from \eqref{eq:vw0}, lead to uniform representations of sizes as in Table~\ref{tab:minnd}.\label{tab:wv}} {\footnotesize \begin{tabular}{ll|ll} \hline \rule{0pt}{2.3ex $n$ & $d$ & $V_1$ & $W_1$\\ \hline \rule{0pt}{2.3ex 3 & 4 & $-$ & $-$\\ 3 & 5 & $-$ & $xy$\\ 3 & 6 & $-$ & $xy,\, x^2y$\\ 3 & 7 & $-$ & $x^2y,\, y^2z,\, z^2 x$\\ 3 & 8 & $-$ & $x^2y,\, y^2z,\, z^2x,\, x^2y^2,\, z^2w^2$ \\ 3 & 9 & $x^3y,\,y^3z,\,z^3x$ & $x^2y,\,x^2z,\,y^2z,\,x^2y^2,\,x^2z^2,\,y^2z^2$\\[2mm] 4 & 4 & $-$ & $xy$\\ 4 & 5 & $-$ & $xy,\, zw$\\ 4 & 6 & $x^2y,\, y^2z,\, z^2w$ & $xy,\, x^2y,\, z w$\\ 4 & 7 & $x^2y,\, y^2x,\, z^2w,\, w^2x,\, xy$ & $x^2z,\, xz^2,\,y^2w,\, yw^2$\\ 4 & 8 & $x^2y,\, x^2y^2,\, z^2x,\, x^3 y,\, y^3z,\, z^3w,\,w^3x$ & $ xy,\,xyz,\, xyw,\, y^2z,\, z^2w,\, w^2x,\, w^2y,\, x^2z$\\[2mm] 5 & 4 & $-$ & $ xy,\, zw$\\ 5 & 5 & $xy,\, yz,\, zw$ & $wu,\, xu$\\[2mm] 6 & 4 & $-$ & $xy,\, zw,\, uv$\\ 6 & 5 & $xy,\, zw,\,uv,\,wy$ & $yz,\, wu,\, xv,\, xz$\\[2mm] 7 & 4 & $-$ & $xy,\, zw,\, uv,\, xq,\, yq$\\ 7 & 5 & $xy,\, z w,\, u v,\, w y,\, q u$ & $y z,\, w u,\, v q,\, x z,\, w x$\\[2mm] 8 & 4 & $-$ & $x y,\, y z,\, x z,\, w u,\, w v,\, u v,\, q s$ \\ \hline \end{tabular}} \end{center} \end{table} \begin{ex} \label{ex:turan} {\rm To show how things get complicated, let us consider the construction for $d=4$. We take $V= \langle 1,x_1,\ldots,x_n,x_1^2,\ldots,x_n^2 \rangle$ and \[W= \langle 1,x_1,\ldots,x_n,x_{\alpha_1}x_{\beta_1},\ldots,x_{\alpha_m}x_{\beta_m} \rangle,\] where $1\le \alpha_i<\beta_i\le n$ and $m$ is as small as possible. If we take all possible pairs $x_{\alpha}x_\beta$, then clearly $F_1 \cdot V \cdot W \supseteq F_4$, while on the other hand, when $m=0$, $F_1 \cdot V \cdot W$ does not contain any monomials of the form \begin{equation} \label{eq:ijkl} x_i \, x_j \, x_k \, x_\ell \end{equation} for $1\le i<j<k<\ell\le n$. We need a minimal set of $x_\alpha x_\beta$ to cover all possible monomials \eqref{eq:ijkl}, which is related to the following covering problem. Given positive integers $r\le k\le n$, we say that a system $S$ of $r$-subsets (called blocks) of $\{1,\dots, n\}$ is called a Tur\'an $(n,k,r)$-system if every $k$-subset of $\{1,\dots, n\}$ contains at least one block from $S$ \cite{Sidorenko}. The minimum size of $S$ is called the Tur\'an number $T(n,k,r)$. In our case, additional terms $x_{\alpha_1}x_{\beta_1},\ldots,x_{\alpha_m}x_{\beta_m}$ form a Tur\'an $(n,4,2)$-system. While for most cases only upper and lower bounds for $T(n,k,r)$ are known, Tur\'an proved that \begin{equation}\label{eq:Turan} T(n,4,2)=mn-3 \, {m(m+1)\over 2}, \end{equation} where $m=\lfloor n/3\rfloor$. To obtain the minimal set one has to divide $\{1,\ldots,n\}$ into three nearly equal groups (their sizes do not differ for more than one) and then take all pairs $x_\alpha x_\beta$ such that $\alpha$ and $\beta$ belong to the same group. As a result, such construction gives a uniform representation of size $N$, where $N=\frac{1}{6} \, n^2+O(n)$, which therefore implies $N^*(n,4) \le \frac{1}{6} \, n^2+O(n)$. \hfill $\clubsuit$ } \end{ex} \section{Matrix polynomials} \label{sec:MatPol} Suppose that we have a uniform representation $M$ of $p_{n,d}$ as in \eqref{eq:polnd}, and write \begin{equation}\label{eq:linM} M=M_0+M_1=M_0 + \sum_{|\alpha| \le d} c_\alpha M_\alpha, \end{equation} where each $M_\alpha \in F_1^{N \times N}$. Now consider the matrix polynomial (cf.~\eqref{eq:polnd}) \[ P_{n,d}=\sum_{|\alpha| \le d} x^\alpha C_\alpha , \] where $C_\alpha$ is a $k\times k$ matrix. We will show that under certain assumptions we can construct from $M$ a matrix $\widetilde{M}$ that represents $P_{n,d}$ in the sense that $\det(\widetilde M)=\det(P_{n,d})$. We obtain $\widetilde M$ from $M$ in the following way. Each element of the form $\alpha +\beta x +\gamma y$ is replaced by the $k\times k$ matrix $(\alpha+\beta x + \gamma y)I_k$, where $I_k$ is the $k \times k$ identity, and each $c_\alpha$ is replaced by the matrix $C_\alpha$. \begin{thm}\label{thm:matpollin} Let \eqref{eq:linM} be a uniform representation of the generic polynomial \eqref{eq:polnd} of degree $d$ in $n$ variables and assume that there exist matrices $Q$ and $Z$, whose elements are polynomials in $x_1,\ldots,x_n$, such that $\det(Q) = \det(Z) = 1$, and $QMZ$ is a triangular matrix with one diagonal element equal to $p_{n,d}$ and all other diagonal elements equal to 1. Then \[\widetilde M = M_0\otimes I_k+ \sum_{|\alpha|\le d}M_\alpha\otimes C_\alpha\] is a representation for the matrix polynomial $P_{n,d}$, i.e., $\det(\widetilde M)=\det(P_{n,d})$. \end{thm} \begin{proof} It is easy to see that $(Q\otimes I_k)\widetilde M(Z\otimes I_k)$ is a block triangular matrix with one diagonal block $P_{n,d}$ while all other diagonal blocks are equal to $I_k$. Since $\det(Q\otimes I_k)=\det(Z\otimes I_k)=1$, it follows that $\det(\widetilde M)=\det(P_{n,d})$. \end{proof} \begin{ex} {\rm Theorem~\ref{thm:matpollin} applies to the uniform representation~\eqref{eq:repjan}. Indeed, take \[ Q=\begin{bmatrix} 1 & & & & & & &\cr & \ddots & & & & & & \cr & & 1 & & & & & \cr & & & 1 & y& y^2 &y^3 &y^4 \cr & & & & 1 & y & y^2& y^3\cr & & & & & 1 & y& y^2\cr & & & & & & 1 & y\cr & & & & & & & 1\end{bmatrix}, \qquad Z=\begin{bmatrix} 1 & & & & & & & \cr x & 1 & & & & & & \cr x^2 & x & 1 & & & & & \cr x^3 & x^2 & x& 1 & & & & \cr x^4 & x^3 & x^2 & x & 1 & & & \cr & & & & & 1 & & \cr & & & & & & \ddots & \cr & & & & & & & 1\end{bmatrix}, \] then \[ QMZ= \begin{bmatrix} & 1 & & & & & & & \\ & & 1 & & & & & & \\ & & & 1 & & & & & \\ & & & & 1 & & & & \\ p & \times & \times & \times &c_{40}& & & & \\ \times &\times &\times &c_{31}& & 1 & & & \\ \times &\times &c_{22}& & & & 1 & & \\ \times &c_{13}& & & & & & 1 & \\ c_{04}& & & & & & & & 1 \end{bmatrix}. \] It is easy to see that there exist permutation matrices $P_L$ and $P_R$ such that \[ P_L(QMZ)P_R= \begin{bmatrix} 1 & & & & & & & & \\ & 1 & & & & & & & \\ & & 1 & & & & & & \\ & & & 1 & & & & & \\ c_{40} & \times & \times & \times &p& & & & \\ &\times &\times &c_{31}& \times & 1 & & & \\ &\times &c_{22}& & \times & & 1 & & \\ &c_{13}& & & \times & & & 1 & \\ & & & & c_{04} & & & & 1 \end{bmatrix} \] is triangular and has the diagonal which satisfies Theorem \ref{thm:matpollin}. Therefore, we can apply \eqref{eq:repjan} for matrix polynomials by using block matrices. This can be generalised to a uniform representation of size $2d+1$ of the form \eqref{eq:repjan}. In a similar way we can show that this also holds for representations of the form \eqref{eq:repjan2} of size $2d-1$. \hfill $\clubsuit$ } \end{ex} Unfortunately, not all uniform determinantal representations induce a determinantal representation of a general matrix polynomial in this manner. As a counterexample, let $M$ be such a uniform determinantal representation of the polynomial $p_{n,d}$, $|\alpha|,|\beta| \le d$, and construct a representation of larger size \[ M'=\begin{bmatrix} M & 0\\ 0 & N\\ \end{bmatrix}, \quad\quad \hbox{with}\quad N= \left[ \begin{array}{rrrr} 0 & c_{\alpha} & c_{\beta} & 1\\ -c_{\alpha} & 0 & 1 & 0\\ -c_{\beta} & -1 & 0 & 0\\ -1 & 0 & 0 & 1 \end{array} \right]. \] Then $\det(M')=\det(M)\det(N)= p_{n,d}(1+c_{\alpha}c_{\beta}-c_{\beta}c_{\alpha})=p_{n,d}$, but $\widetilde{M'}$ is not a representation for the matrix polynomial $P_{n,d}$ as the coefficient matrices $C_{\alpha}$ and $C_{\beta}$ do not commute in general. This motivates the following definition. \begin{de} A uniform determinantal representation $M$ is \emph{minimal} if there do not exist constant matrices $P$ and $Z$ such that $\det(P)=\det(Z)=1$ and \[ QMZ=\begin{bmatrix} * & *\\ 0 & M_2\\ \end{bmatrix}, \quad \quad \hbox{where $M_2$ is square with} \quad \det(M_2) =1. \] \end{de} We speculate that each minimal uniform representation gives rise to a representation for a matrix polynomial. \section{Numerical experiments} \label{sec:Numerics} Recently, a new numerical approach for computing roots of systems of bivariate polynomials was proposed in \cite{BorMichiel}. The main idea is to treat the system as a two-parameter eigenvalue problem using determinantal representations. Suppose that we are looking for roots of a system of bivariate polynomials \begin{equation}\label{eq:syspol} \begin{matrix}p=\sum_{i+j \le d_1} \alpha_{ij} x^i y^j=0,\\[0.2em] q=\sum_{i+j \le d_2} \beta_{ij} x^i y^j=0,\end{matrix} \end{equation} where $p$ and $q$ are polynomials of degree $d_1$ and $d_2$ over $\CC$. Let $P=A_0 + x A_1 + y A_2$ and $Q=B_0 + x B_1 + y B_2$, where $A_0,A_1,A_2 \in \CC^{N_1 \times N_1}$ and $B_0,B_1,B_2 \in \CC^{N_2 \times N_2}$, with $\det(P)=p$ and $\det(Q)=q$, be determinantal representations of $p$ and $q$, respectively. Then a root $(x,y)$ of \eqref{eq:syspol} is an eigenvalue of the two-parameter eigenvalue problem \begin{equation}\label{eq:twopar} \begin{matrix} (A_0+x A_1+y A_2) \, u=0,\\[0.2em] (B_0+x B_1+y B_2) \, v=0, \end{matrix} \end{equation} where $u\in\CC^{N_1}$ and $v\in\CC^{N_2}$ are nonzero vectors. The standard way to solve \eqref{eq:twopar} is to consider a joint pair of generalized eigenvalue problems \cite{Atkinson} \begin{equation}\label{eq:twopardelta} \begin{matrix} (\Delta_1-x \Delta_0) \, w=0,\\[0.2em] (\Delta_2-y \Delta_0) \, w=0, \end{matrix} \end{equation} where \[ \Delta_0=A_1\otimes B_2-A_2\otimes B_1,\quad \Delta_1=A_2\otimes B_0-A_0\otimes B_2,\quad \Delta_2=A_0\otimes B_1-A_1\otimes B_0, \] and $w=u\otimes v$. In this particular application we can expect that the pencils in \eqref{eq:twopardelta} are singular, i.e., $\det(\Delta_1-x\Delta_0)\equiv 0$ and $\det(\Delta_1-y\Delta_0)\equiv 0$. Namely, by B\'ezout's theorem a generic system \eqref{eq:syspol} has $d_1d_2$ solutions, while a generic problem \eqref{eq:twopar} has $N_1N_2$ eigenvalues. Unless $(d_1,d_2)=(N_1,N_2)$, both pencils in \eqref{eq:twopardelta} are singular. In this case we first have to apply the staircase algorithm from \cite{MuhicPlestenjakLAA} to extract the finite regular eigenvalues. The method returns smaller matrices $\widetilde \Delta_0$, $\widetilde \Delta_1$, and $\widetilde \Delta_2$ (of size $d_1d_2\times d_1d_2$ for a generic \eqref{eq:syspol}) such that $\widetilde \Delta_0$ is nonsingular and ${\widetilde \Delta_0}^{-1}\widetilde \Delta_1$ and ${\widetilde \Delta_0}^{-1}\widetilde \Delta_2$ commute. From \[ \begin{matrix} (\widetilde\Delta_1-x \widetilde\Delta_0) \, \widetilde w=0,\\[0.2em] (\widetilde\Delta_2-y \widetilde\Delta_0) \, \widetilde w=0, \end{matrix} \] we compute the eigenvalues $(x,y)$ using a variant of the QZ algorithm \cite{HKP} and thus obtain the roots of \eqref{eq:syspol}. The above approach is implemented in the Matlab package BiRoots \cite{BorBR} together with the two determinantal representations from \cite{BorMichiel}. The first one, to which we refer as {\tt Lin1}, is a uniform one from Example \ref{ex:lin1} of size $\frac{1}{4} \, d^2+ O(d)$ for a polynomial of degree $d$. The second one, which we refer to as {\tt Lin2}, is not uniform and involves some computation to obtain a smaller size $\frac{1}{6} \, d^2+ O(d)$. Although the construction of {\tt Lin2} is more time consuming, this pays off later, when the staircase algorithm is applied to \eqref{eq:twopardelta}. Table~\ref{tbl:compare1} shows the sizes of determinantal representations for polynomials of small degree. As expected, the new uniform determinantal representation of size $2d-1$, to which we refer as {\tt MinUnif}, returns smaller matrices, which reflects later in faster computational times. It is also important that {\tt Lin1} and {\tt MinUnif} return real matrices for polynomials with real coefficients, which is not true for {\tt Lin2}. \begin{table}[!htbp] \begin{footnotesize} \begin{center} \caption{Size of the matrices for {\tt Lin1} and {\tt Lin2} for bivariate polynomials ($n=2$) and various degrees $d$.}\label{tbl:compare1} \begin{tabular}{l|cccccccccc} \hline \rule{0pt}{2.3ex Method & $d=3$ & $d=4$ & $d=5$ & $d=6$ & $d=7$ & $d=8$ & $d=9$ & $d=10$ & $d=11$ & $d=12$\\ \hline \rule{0pt}{2.3ex {\tt Lin1} & 5 & 8 & 11 & 15 & 19 & 24 & 29 & 35 & 41 & 48 \\ {\tt Lin2} & 3 & 5 & \ph{1}8 & 10 & 13 & 17 & 20 & 24 & 29 & 34\\ {\tt MinUnif} & 5 & 7 & \ph{1}9 & 11 & 13 & 15 & 17 & 19 & 21 & 23\\ \hline \end{tabular} \end{center} \end{footnotesize} \end{table} It was reported in \cite{BorMichiel} that the determinantal representation approach for solving systems of bivariate polynomials is competitive for polynomials of degree 9 or less. As we show below, the new uniform representation {\tt MinUnif} extends this to degree 15 and, in addition, performs better than the existing representations for polynomials of degree 6 or more. In \cite{BorMichiel} the approach was compared numerically to the following state-of-the art numerical methods for polynomial systems: {\tt NSolve} in Mathematica~9~\cite{Wolfram}, BertiniLab~1.4~\cite{Bertini} running Bertini~1.5~\cite{BertiniExe}, NAClab~3.0~\cite{NAClab}, and {\tt PHCLab}~1.04~\cite{PHClab} running PHCpack~2.3.84, which turned out as the fastest of these methods. To show the improved performance of the new determinantal representation, we compare {\tt MinUnif} to {\tt Lin1}, {\tt Lin2}, and {\tt PHCLab} in Table~\ref{tbl:two}. For each $d$ we run the methods on the same set of 50 real and 50 complex random polynomial systems of degree $d$ and measure the average time. For {\tt Lin1} and {\tt MinUnif}, where determinantal representations have real matrices for real polynomials, we report separate results for polynomials with real and complex coefficients. The timings for {\tt Lin1} and {\tt Lin2} are given only for $n\le 10$ as for larger $n$ these two linearisations are no longer competitive. \begin{table}[!htbp] \begin{footnotesize} \begin{center} \caption{Average computational times in milliseconds for {\tt Lin1}, {\tt Lin2}, {\tt MinUnif}, and {\tt PHCLab} for random full bivariate polynomial systems of degree $3$ to $15$. For {\tt Lin1} and {\tt MinUnif} separate results are included for real $(\RR)$ and complex polynomials $(\CC)$. }\label{tbl:two} \begin{tabular}{r|rrrrrr} \hline \rule{0pt}{2.3ex $d$ & {\tt Lin1} ($\RR$) & {\tt Lin1} ($\CC$) & {\tt Lin2} & {\tt PHCLab} & {\tt MinUnif} ($\RR$) & {\tt MinUnif} ($\CC$) \\ \hline \rule{0pt}{2.3ex 3 & \ph{111}6 & \ph{111}8 & \ph{11}4 & \ph{1}116 & \ph{11}6 & \ph{11}7 \\ 4 & \ph{111}9 & \ph{11}11 & \ph{11}6 & \ph{1}130 & \ph{1}12 & \ph{11}13 \\ 5 & \ph{11}20 & \ph{11}26 & \ph{1}13 & \ph{1}151 & \ph{1}18 & \ph{11}20 \\ 6 & \ph{11}39 & \ph{11}71 & \ph{1}28 & \ph{1}174 & \ph{1}27 & \ph{11}27 \\ 7 & \ph{11}96 & \ph{1}160 & \ph{1}51 & \ph{1}217 & \ph{1}36 & \ph{11}44 \\ 8 & \ph{1}205 & \ph{1}395 & 118 & \ph{1}264 & \ph{1}59 & \ph{11}74 \\ 9 & \ph{1}467 & 1124 & {279} & \ph{1}329 & \ph{1}{95} & \ph{1}125 \\ 10 & 1424 & 3412 & 600 & \ph{1}414 & {147} & \ph{1}221 \\ 11 & & & & \ph{1}538 & {248} & \ph{1}354 \\ 12 & & & & \ph{1}650 & {361} & \ph{1}530 \\ 13 & & & & \ph{1}911 & {592} & \ph{1}740 \\ 14 & & & & 1142 & {842} & 1148 \\ 15 & & & & 1531 & {1237} & 1835 \\ \hline \end{tabular} \end{center} \end{footnotesize} \end{table} Of course, the computational time is not the only important factor, we also have to consider the accuracy and reliability. In each step of the staircase algorithm a rank of a matrix has to be estimated numerically, which is a delicate task. After several steps it may happen that the gap between the important singular values and the meaningful ones that should be zero in exact computation, virtually disappears. In such case the algorithm fails and does not return any roots. As the number of steps in the staircase algorithm increases with degree of the polynomials, such problems occur more often for polynomials of large degree. A heuristic that usually helps in such cases is to apply the algorithm on a transformed system \begin{align*} \widetilde p &:= \ph{-}c p + s q = 0,\\ \widetilde q &:= -s p + c q = 0 \end{align*} for random $c$ and $s$ such that $c^2+s^2=1$. As this transformation does not change the conditioning of the roots, we can conclude that the difficulties with the staircase algorithm are not directly related to the conditioning. The trick does not work every time, and it seems that for some systems the only way to make the determinantal representation approach to work is to increase the machine precision. We can apply the same approach to systems of polynomials in more than two variables. However, since the size of the corresponding $\Delta$ matrices is the product of sizes of all representations, this is competitive only for $n=3$ and $d\le 3$. For a comparison, if we have a system of three polynomials in three variables of degree $3$, then the size of the $\Delta$ matrices is $343\times 343$. For degree $4$ the size increases to $1000\times 1000$ and PHCpack is faster. Finally, for $n=4$ and the smallest nontrivial $d=2$ we already get $\Delta$ matrices of size $625\times 625$ and the method is not efficient. \section{Outlook} \label{sec:Outlook} We have introduced {\em uniform determinantal representations}, which rather than representing a single polynomial as the determinant of a matrix of affine-linear forms, represent all polynomials of degree at most $d$ in $n$ variables as such a determinant. We have seen that in the bivariate case, these determinantal representations are useful for numerically solving bivariate systems of equations; and in the general multivariate case we have determined, up to constants, the asymptotic behaviour of $N^*(n,d)$, the minimal size of such a representation, for $n$ fixed and $d \to \infty$. We now summarise several results that have been shown in the paper. \begin{itemize} \item For fixed $n$ and $d \to \infty$, $N^*(n,d) \sim d^{n/2}$, see Theorem~\ref{thm:Main}. This is a noticeable improvement on \cite{Quarez}, where an asymptotic rate of $N^*(n,d) \sim d^n$ is shown, with the remark that the representation in \cite{Quarez} are symmetric. However, symmetry currently cannot be exploited by methods that compute roots of multivariate polynomial systems. \item For fixed odd $d$ and $n \to \infty$, $N^*(n,d) \sim n^{(d-1)/2}$, which is the same rate as Quarez \cite{Quarez}, who also manages to get symmetric representations. For fixed even $d$ and $n \to \infty$, $N^*(n,d) \lsim n^{d/2}$, which again is the same rate as in \cite{Quarez}. However, we have a slightly smaller lower bound for the asymptotic rate of $n^{(d-1)/2}$. \item Tables~\ref{tab:minnd} and \ref{tab:wv} give constructions for the smallest representations that we have been able to find for some small values of $n$ and $d$. \item $N^*(n,2) \le n+1$; cf.~Table~\ref{tab:minnd}. \item $N^*(n,3) \le 2n+1$; cf.~Table~\ref{tab:minnd}. \item $N^*(n,4) \le \frac{1}{6} n^2+O(n)$; see Example~\ref{ex:turan}. \item $N^*(2,d) \le 2d-1$; cf.~Table~\ref{tab:minnd}. Note that this result satisfies Dixon~\cite{Dixon} up to an asymptotic factor 2 whereby no computations are necessary for the determinantal representation. In particular, it is a major improvement on the $\sim \frac{1}{4} d^2$ of \cite{Quarez, BorMichiel}. \item Due to the smaller sizes of the representations, the numerical approach for bivariate polynomials ($n=2$) is competitive to (say) Mathematica for degree $d$ up to $d \approx 15$ (see Section~\ref{sec:Numerics}); this in contrast to $d \approx 9$ as obtained in \cite{BorMichiel}. \item Under some conditions, the results carry over to the case of matrix coefficients (see Section~\ref{sec:MatPol}). \end{itemize} There are still many interesting open questions, both of intrinsic mathematical interest and of relevance to polynomial system solving. First of all, in a situation where the degree $d$ is fixed and the number $n$ of variables grows, what is the asymptotic behaviour of $N^*(n,d)$? Although we expect that the above inequalities are equalities, whether this holds is still open. For general fixed $d$, the proof of Theorem~\ref{thm:Main} yields a lower bound which is a constant (depending on $d$) times $n^{(d-1)/2}$. For odd $d$ we obtain a matching upper bound (with a different constant) by using Proposition~\ref{prop:Cons2} with $V=W=F_{(d-1)/2}$. However, for even $d$ we only know how to obtain $O(n^{d/2})$. We remark that this latter is (up to a constant) the same bound as obtained in \cite[Thm.~4.4]{Quarez} for {\em symmetric} uniform representations. Second, in the case of fixed $n$ and varying $d$ studied in this paper, what are the best constants in Theorem~\ref{thm:Main}? More specifically, for fixed $n$, does $\lim_{d \to \infty} \frac{N^*(n,d)}{d^{n/2}}$ exist, and if so, what is its value? Third, how can our techniques for upper bounds and lower bounds be further sharpened? Can singular matrix spaces {\em other} than compression spaces be used to obtain tighter upper bounds (constructions) on $N^*(n,d)$? Can the action of the affine group be used more systematically to find lower bounds on $N^*(n,d)$? Fourth, is it true that each minimal uniform representation gives rise to a representation of the corresponding matrix polynomial (cf.~Section~\ref{sec:MatPol})? Finally, we have restricted our attention to matrices that, apart from being affine-linear in $x_1,\dots,x_n$, are also affine-linear in the coefficients $c_\alpha$. Our proofs give the same asymptotic behaviour (with different constants) if we require, in addition, that no quadratic terms $c_\alpha x_i$ may occur in $M$. If, instead, we relax the condition that $M$ be affine-linear in the $c_\alpha$ to a {\em polynomial} dependence on the $c_\alpha$, then the same bounds still apply; see Remark~\ref{re:Poly}. But what if we relax this to {\em rational} dependence of $M$ on the $c_\alpha$? Given that $p_{n,d}$ is only linear in the $c_\alpha$ it seems unlikely that allowing $M$ to be rational in the $c_\alpha$ we would gain anything, but we currently do not know how to formalise this intuition. On the other hand, in cases where a (non-uniform) determinantal representation of size $d$ is known to exist for every (or sufficiently general) polynomials of degree $d$ in $d$ variables (e.g., in the case of plane curves), it follows that this representation can be chosen to have entries {\em algebraic} in the $c_\alpha$. This observation rules out approaches aimed at proving lower bounds in a too general setting. \vskip1in Abstract: The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.

\subsection{Lagrangian for FCNC $Z'$s} \begin{figure}[th!] \begin{center} \includegraphics[width=10cm]{gqtZp.eps} \caption{Leading-order diagrams for $gu\rightarrow tZ'$ with anomalous $t$-$u$-$Z'$ coupling and $gc\rightarrow tZ'$ with anomalous $t$-$c$-$Z'$ coupling.} \label{gqtZp} \end{center} \end{figure} An FCNC term in the Lagrangian that includes the anomalous coupling of a $t,q$ pair to a $Z'$ boson is given by \begin{equation} {\cal L}_{FCNC} = \frac{1}{ \Lambda } \, \kappa_{tqZ'} \, e \, \bar t \, \sigma_{\mu\nu} \, q \, F^{\mu\nu}_{Z'} + h.c., \label{Langrangian} \end{equation} where $\kappa_{tqZ'}$ is the anomalous $t$-$q$-$Z'$ coupling, with $q$ an up or charm quark; $e$ is the electron charge; $\Lambda$ is an effective new physics scale in the few TeV's range; $F^{\mu\nu}_{Z'}$ is the $Z'$ field tensor; and $\sigma_{\mu \nu}=(i/2)(\gamma_{\mu}\gamma_{\nu} -\gamma_{\nu}\gamma_{\mu})$ with $\gamma_{\mu}$ the Dirac matrices. The partonic processes involved are $gu \rightarrow tZ'$ and $gc \rightarrow tZ'$. Leading-order diagrams for these processes are shown in Fig. \ref{gqtZp}. Related processes involving $Z$ bosons with anomalous couplings were studied in Refs. \cite{Kidonakis:2003sc,Kidonakis:2017mfy}. \subsection{\label{stringyZprime} Lagrangian for string-inspired $Z'$s } The Lagrangian for a $Z'$ coming from string-inspired models is given below, where we adopt the notation introduced in Refs.~\cite{Coriano:2008wf,Faraggi:2015iaa}. Here we report the most basic definitions for completeness. The fermion-fermion-$Z^{\prime}$ interaction is given by \begin{eqnarray} \sum_{i=L,R} z_{t,i} g_{Z'} \bar{t}_i \gamma^{\mu} t_i Z_{\mu}^{\prime}, \end{eqnarray} where the coefficients $z_{t,L}$, and $z_{t,R}$ are the charges of the left- and right-handed top quarks respectively. The $Z'$ coupling is indicated by $g_{Z'}$. The mass of the $Z$ gauge boson is parametrized in terms of the vacuum expectation values (vev's) of the Higgs sector $v_{H_1}$, $v_{H_2}$ as follows \begin{eqnarray} &&m_Z^2=\frac{g^2}{4 \cos^2\theta_W}(v_{H_1}^2+v_{H_2}^2)\left[1+O(\varepsilon^2)\right] \, , \nonumber\\ &&\varepsilon=\frac{\delta m^2_{Z Z^{\prime}}}{m^2_{Z^{\prime}}-m^2_{Z}} \, , \nonumber\\ &&\delta m^2_{Z Z^{\prime}}=-\frac{g g_{Z'}}{4\cos\theta_W}(z_{H_1}^2 v_{H_1}^2+z_{H_2}^2v_{H_2}^2) \, , \end{eqnarray} where the mixing parameter $\varepsilon$ is defined perturbatively, $z_{H_1}$ and $z_{H_2}$ are the charges of the Higges, $g=e/\sin\theta_W$, $g_Y= e/\cos\theta_W$, and $\theta_W$ is the Weinberg angle. We consider $m_{Z'}$ as a free parameter in the TeV's range. We restrict our attention to the interaction Lagrangian for the top-quark sector only, which is written as \begin{eqnarray} &&{\mathcal{L}}_{int}= \bar{t}_{L} N^{Z^{\prime}}_{L}\gamma^{\mu}t_{L} Z^{\prime}_{\mu} +\bar{t}_{R}N^{Z^{\prime}}_{R}\gamma^{\mu}t_{R} Z^{\prime}_{\mu}\,, \end{eqnarray} where the left-handed (L) and right-handed (R) couplings are \begin{eqnarray} &&N^{Z^{\prime}}_{L}=-i\left(-g\cos\theta_W T_{3,L}\varepsilon +g_Y\sin\theta_W \frac{ Y_{t,L}}{2}\varepsilon+g_{Z'}\frac{ z_{t,L}}{2}\right) \, , \nonumber\\ &&N^{Z'}_{R}=-i\left( g_Y \sin\theta_W \frac{ Y_{t,R}}{2}\varepsilon +g_{Z'} \frac{z_{t,R}}{2}\right) \,, \end{eqnarray} where $Y_{t,L/R}$ is the hypercharge and $T_{3,L}$ is the weak isospin. \begin{figure}[th!] \begin{center} \includegraphics[width=10cm]{gttZp.eps} \caption{Leading-order diagrams for $gt\rightarrow tZ'$.} \label{gttZp} \end{center} \end{figure} Based on this Lagrangian, we will study below the process $gt \rightarrow tZ'$. The leading-order diagrams for this process are shown in Fig. \ref{gttZp}. \subsection{Hadronic cross section} The hadronic cross section for $p(P_1)+p(P_2)\rightarrow t(p_t)+Z'(p_{Z'})$ is expressed in terms of Mandelstam variables \begin{eqnarray} S=(P_1+P_2)^2\,, ~~T=(P_1-p_t)^2 \,, ~~U=(P_2-p_t)^2 \,, ~~S_4 = S + T+ U-m_t^2-m_{Z'}^2\,. \end{eqnarray} We also define $T_1=T -m_t^2$ and $U_1=U -m_t^2$. The factorized differential cross section can be written as \begin{eqnarray} S^2\frac{d^2 \sigma(S,T_1,U_1)}{dT_1 ~dU_1} &=& \sum_{i,j=q,g} \int_{x_1^-}^{1} \frac{dx_1}{x_1} \int_{x_2^-}^{1}\frac{dx_2}{x_2} f_{i/p_1}(x_1,\mu_F^2) f_{j/p_2}(x_2,\mu_F^2) \nonumber\\ &&\times \, \hat{\sigma}_{ij\rightarrow tZ'}(s,t_1,u_1,m_t^2,m^2_{Z'},\mu_F^2,\alpha_s(\mu_R^2)) + {\cal O}(\Lambda_{QCD}^2/\Lambda^2) \label{diffXsec} \end{eqnarray} where $f_{j/p}(x,\mu_F^2)$ is the parton distribution function representing the probability of finding the parton $j$ in proton $p$, $\mu_F$ and $\mu_R$ are the factorization and renormalization scales respectively, and $\hat\sigma_{ij\rightarrow tZ'}$ is the hard scattering cross section. Here, $\Lambda_{QCD}$ is the QCD scale while the scale $\Lambda$ is of the order of $m_{Z'}$, and power suppressed terms $\Lambda_{QCD}^2/\Lambda^2$ are neglected. In our numerical results in Sec. 4 we set $\mu_F=\mu_R=\mu$. The lower integration limits in the factorization formula are given by \begin{equation} x_1^-= -\frac{U_1}{S + T_1} \,, ~~ x_2^-= -\frac{x_1 T_1}{x_1 S + U_1}. \end{equation} The double-differential cross section in Eq. (\ref{diffXsec}) can be written in terms of the transverse momentum $p_T$ of the top quark and its rapidity $y$ using \begin{equation} T_1= - \sqrt{S} ~m_T e^{-y}\,,~~ U_1= - \sqrt{S} ~m_T e^y\,, \end{equation} where the transverse mass $m_T$ is defined as $m_T=\sqrt{p_T^2 + m_t^2}$. \subsection{Leading-order cross sections} For the partonic process $g(p_g)+q(p_q) \rightarrow t(p_t)+Z'(p_{Z'})$, we define the kinematical variables $s=(p_g+p_q)^2$, $t=(p_g-p_t)^2$, and $u=(p_q-p_t)^2$. The leading-order (LO) double-differential partonic cross section for $g q \rightarrow tZ'$, with $q$ and up or charm quark, via anomalous couplings is \begin{equation} \frac{d^2{\hat\sigma^{(0)}}_{gq \rightarrow t Z'}}{dt \, du} =F^{\rm LO}_{gq \rightarrow t Z'} \, \delta(s_4) \, , \label{LO} \end{equation} where \begin{eqnarray} F^{\rm LO}_{gq \rightarrow t Z'}&=& \frac{2 \pi \alpha \alpha_s \kappa_{tqZ'}^2}{3s^3 (t-m_t^2)^2 \Lambda^2} \left\{2 m_t^8 -m_t^6(3 m_{Z'}^2+4 s+2 t) \right. \nonumber \\ && {}+m_t^4\left[2m_{Z'}^4-m_{Z'}^2(2s+t)+2(s^2+4st+t^2)\right] \nonumber \\ && {}+m_t^2\left[2m_{Z'}^6-4m_{Z'}^4t+m_{Z'}^2(s+t)(s+5t) -2t(3s^2+6st+t^2)\right] \nonumber \\ && \left. {}-t\left[2 m_{Z'}^6-2m_{Z'}^4(s+t) +m_{Z'}^2(s+t)^2-4 s t(s+t)\right]\right\} \, , \end{eqnarray} with $\alpha=e^2/(4\pi)$. For the partonic process $g(p_g)+t(p_q) \rightarrow t(p_t)+Z'(p_{Z'})$, we again define the kinematical variables $s=(p_g+p_q)^2$, $t=(p_g-p_t)^2$, and $u=(p_q-p_t)^2$. The LO cross section for $g t \rightarrow t Z'$ is given by \begin{eqnarray} &&F^{\rm LO}_{g t\rightarrow t Z'}= 4 \frac{4\pi \alpha_s}{m_{Z'}^2 N_c (s-m_t^2)^2 (t-m_t^2)^2} \left\{g_{AtZ'}^2 \left[2 m_{Z'}^4 ((2m_t^2 s t-5 m_t^4 (s+t)+6 m_t^6)+s t(s+t)) \right.\right. \nonumber\\ &&\left.\left. +2 m_{Z'}^6 (s-m_t^2)(m_t^2-t)-m_{Z'}^2 (m_t^4 (s-3 t) (3 s-t) -12 m_t^6 (s+t)-m_t^2 (s+t) (-6 st+s^2+t^2) \right.\right. \nonumber\\ &&\left.\left. +18 m_t^8+s t (s^2+t^2))-2 m_t^2 (s-m_t^2) (t-m_t^2) (-2 m_t^2+s+t)^2\right] \right. \nonumber\\ &&\left. +g_{VtZ'}^2 m_{Z'}^2 \left[-s t (2 m_{Z'}^4+s^2+t^2-2 m_{Z'}^2 (s+t)) -m_t^4 (-2 m_{Z'}^2 (s+t)+2 m_{Z'}^4+14 st+3 s^2+3 t^2) \right.\right. \nonumber\\ &&\left.\left. +m_t^2 (-8 m_{Z'}^2 s t+2 m_{Z'}^4 (s+t)+(s+t) (6 s t+s^2+t^2))+6 m_t^8\right]\right\} \,, \nonumber\\ \end{eqnarray} where the vector and axial coupling of the $Z^{\prime}$ boson to the top quark are \begin{eqnarray} &&\frac{-i g}{4 c_w}\gamma^{\mu} g_{VtZ^{\prime}}=\frac{-i g}{c_w} \frac{1}{2}\left[ -\varepsilon c_w^2 T_3^{L} +\varepsilon s_w^2(\frac{Y_{t,L}}{2}+\frac{Y_{t,R}}{2}) +\frac{g_{Z'}}{g}c_w(\frac{z_{t,L}}{2}+ \frac{z_{t,R}}{2})\right]\gamma^{\mu} \nonumber\\ &&\frac{-i g}{4 c_w}\gamma^{\mu}\gamma^{5} g_{AtZ^{\prime}}=\frac{-i g}{c_w} \frac{1}{2}\left[ \varepsilon c_w^2 T_3^{L} +\varepsilon s_w^2(\frac{Y_{t,R}}{2}-\frac{Y_{t,L}}{2}) +\frac{g_{Z'}}{g}c_w(\frac{z_{t,R}}{2}- \frac{z_{t,L}}{2})\right]\gamma^{\mu}\gamma^{5}, \nonumber\\ \end{eqnarray} where we set $\sin\theta_W=s_w$ and $\cos\theta_W=c_w$ for brevity. \setcounter{equation}{0}\section{\label{Soft-gluons} Soft-gluon corrections} We next describe the formalism and procedure for calculating soft-gluon corrections in the cross section for $tZ'$ production. For the processes $gq \rightarrow tZ'$ and $gt \rightarrow tZ'$, we defined the usual kinematical variables $s$, $t$, and $u$, in the previous section. We can also define a threshold kinematical variable, $s_4=s+t+u-m_t^2-m_{Z'}^2$, that measures distance from partonic threshold, and vanishes at partonic threshold where there is no energy available for additional radiation. More specifically, $s_4$ is the squared invariant mass of additional final-state radiation. We also define $t_1=t-m_t^2$, $t_2=t-m_{Z'}^2$, $u_1=u-m_t^2$, and $u_2=u-m_{Z'}^2$. The resummation of soft-gluon contributions to the partonic process follows from the factorization of the cross section as a product of functions that describe soft and collinear emission. Taking the Laplace transform ${\hat \sigma}(N)=\int (ds_4/s) \; e^{-N s_4/s} {\hat \sigma}(s_4)$, we have a factorized expression in $4-\epsilon$ dimensions, \begin{equation} \frac{d^2{\hat \sigma}_{gq \rightarrow tZ'}(N,\epsilon)}{dt \, du}= H_{gq \rightarrow tZ'} \left(\alpha_s(\mu)\right)\; S_{gq \rightarrow tZ'} \left(\frac{m_t}{N \mu},\alpha_s(\mu) \right)\; \prod_{i=g,q} J_i\left (N,\mu,\epsilon \right) \label{factsigma} \end{equation} where $H_{gq \rightarrow tZ'}$ is a hard function, $S_{gq \rightarrow tZ'}$ is a soft function for noncollinear soft-gluon emission, and $J_i$ are jet functions for soft and collinear emission from the incoming quark and gluon. Our considerations are identical for all three processes to be studied in this paper, i.e. $gu\rightarrow tZ'$, $gc \rightarrow tZ'$, and $gt \rightarrow tZ'$. The dependence of the soft function $S_{gq \rightarrow tZ}$ on $N$ is resummed via renormalization group evolution \cite{Kidonakis:2017mfy,Forslund:2018qcp,NKttbar,NKsingletop,Kidonakis:1997gm}, \begin{equation} S^b_{gq \rightarrow tZ'}=(Z^S)^* \; S_{gq \rightarrow tZ'} \, Z^S \, , \end{equation} with $S^b_{gq \rightarrow tZ}$ the unrenormalized quantity and $Z^S$ a renormalization constant. The function $S_{gq \rightarrow tZ}$ obeys the renormalization group equation \begin{equation} \left(\mu \frac{\partial}{\partial \mu} +\beta(g_s, \epsilon)\frac{\partial}{\partial g_s}\right)\,S_{gq \rightarrow tZ'} =-2 \, S_{gq \rightarrow tZ'} \, \Gamma^S_{gq \rightarrow tZ'} \, , \end{equation} where $g_s^2=4\pi\alpha_s$, $\beta(g_s, \epsilon)=-g_s \epsilon/2 + \beta(g_s)$ with $\beta(g_s)$ the QCD beta function, and \begin{equation} \Gamma^S_{gq \rightarrow tZ'}=\frac{dZ^S}{d\ln\mu} (Z^S)^{-1} =\beta(g_s, \epsilon) \frac{\partial Z^S}{\partial g_s} (Z^S)^{-1} \end{equation} is the soft anomalous dimension that determines the evolution of $S_{gq \rightarrow tZ}$. The soft anomalous dimension $\Gamma^S_{gq \rightarrow tZ}$ is calculated in dimensional regularization from the coefficients of the ultraviolet poles of the loop diagrams involved in the process \cite{Kidonakis:2017mfy,Forslund:2018qcp,NKttbar,NKsingletop,Kidonakis:2018ybz,Kidonakis:1997gm,Kidonakis:2009ev,Kidonakis:2019nqa}. The resummed partonic cross section in moment space is then given by \begin{eqnarray} \frac{d^2{\hat{\sigma}}^{\rm resum}_{gq \rightarrow tZ}(N)}{dt \, du} &=& \exp\left[\sum_{i=g,q} E_i(N_i)\right] H_{gq \rightarrow tZ'} \left(\alpha_s(\sqrt{s})\right) \; S_{gq \rightarrow tZ'}\left(\alpha_s(\sqrt{s}/{\tilde N'}) \right) \nonumber \\ && \times \exp \left[2\int_{\sqrt{s}}^{{\sqrt{s}}/{\tilde N'}} \frac{d\mu}{\mu}\; \Gamma^S_{gq \rightarrow tZ'} \left(\alpha_s(\mu)\right)\right] \, . \label{resum} \end{eqnarray} Soft-gluon resummation is the exponentiation of logarithms of $N$. The first exponent in Eq. (\ref{resum}) includes soft and collinear corrections \cite{Sterman:1986aj, Catani:1989ne} from the incoming partons, and can be found explicitly in \cite{NKsingletop}. We write the perturbative series for the soft anomalous dimension for $gq \rightarrow tZ'$ as $\Gamma^S_{gq \rightarrow tZ'}=\sum_{n=1}^{\infty}(\alpha_s/\pi)^n \Gamma^{S \, (n)}_{gq \rightarrow tZ'}$. To achieve resummation at next-to-leading-logarithm (NLL) accuracy we require the one-loop result which is given, in Feynman gauge, by \begin{equation} \Gamma^{S\, (1)}_{gq \rightarrow tZ}= C_F \left[\ln\left(\frac{-u_1}{m_t\sqrt{s}}\right) -\frac{1}{2}\right] +\frac{C_A}{2} \ln\left(\frac{t_1}{u_1}\right) \, , \label{tZ1l} \end{equation} with color factors $C_F=(N_c^2-1)/(2N_c)$ and $C_A=N_c$, where $N_c=3$ is the number of colors. Upon expanding the resummed cross section to fixed order and inverting from the transform moment space back to momentum space, the logarithms of $N$ produce ``plus'' distributions of logarithms of $s_4/m_{Z'}^2$. The highest power of these logarithms is 1 at NLO and 3 at NNLO. The NLO soft-gluon corrections for $g q \rightarrow tZ'$ are \begin{eqnarray} \frac{d^2{\hat\sigma}^{(1)}_{gq\rightarrow t Z'}}{dt \, du} &=&F^{\rm LO}_{gq \rightarrow t Z'} \frac{\alpha_s(\mu_R^2)}{\pi} \left\{ 2 (C_F+C_A) \left[\frac{\ln(s_4/m_{Z'}^2)}{s_4}\right]_+ \right. \nonumber \\ && \hspace{-23mm} {}+\left[2 C_F \ln\left(\frac{u_1}{t_2}\right) +C_F \ln\left(\frac{m_{Z'}^2}{m_t^2}\right)-C_F +C_A \ln\left(\frac{t_1}{u_1}\right) +C_A \ln\left(\frac{s m_{Z'}^2}{u_2^2}\right) \right. \nonumber \\ && \hspace{-15mm} \left. {}-(C_F+C_A)\ln\left(\frac{\mu_F^2}{m_{Z'}^2}\right)\right] \left[\frac{1}{s_4}\right]_+ \nonumber \\ && \hspace{-23mm} \left. {}+\left[\left(C_F \ln\left(\frac{-t_2}{m_{Z'}^2}\right) +C_A \ln\left(\frac{-u_2}{m_{Z'}^2}\right) -\frac{3}{4}C_F\right)\ln\left(\frac{\mu_F^2}{m_{Z'}^2}\right) -\frac{\beta_0}{4}\ln\left(\frac{\mu_F^2}{\mu_R^2}\right)\right] \delta(s_4)\right\} \, , \label{NLOgqtZp} \end{eqnarray} where $\beta_0=(11C_A-2n_f)/3$ is the lowest-order QCD $\beta$ function, with $n_f$ the number of light quark flavors. We set $n_f=5$ for $gu\rightarrow tZ'$ and $gc\rightarrow tZ'$, and $n_f=6$ for $gt\rightarrow tZ'$. The leading logarithms in the NLO expansion are the $[\ln(s_4/m_{Z'}^2)/s_4]_+$ terms while the NLL are the $[1/s_4]_+$ terms. In addition, at NLL we determine in Eq. (\ref{NLOgqtZp}) the $\delta(s_4)$ terms involving the scale. In top-quark production processes, the NLO soft-gluon corrections approximate very well the complete NLO corrections \cite{Kidonakis:2017mfy,Forslund:2018qcp,NKttbar,NKsingletop,Kidonakis:2018ybz}. We denote the sum of the LO cross section and the NLO soft-gluon corrections as approximate NLO (aNLO). The NNLO soft-gluon corrections for $g q \rightarrow tZ'$ are \begin{eqnarray} \frac{d^2{\hat\sigma}^{(2)}_{gq\rightarrow t Z'}}{dt \, du} &=&F^{\rm LO}_{gq \rightarrow t Z'} \frac{\alpha_s^2(\mu_R^2)}{\pi^2} \left\{ 2(C_F+C_A)^2 \left[\frac{\ln^3(s_4/m_{Z'}^2)}{s_4}\right]_+ \right. \nonumber \\ && \hspace{-25mm} {}+3(C_F+C_A)\left[2 C_F \ln\left(\frac{u_1}{t_2}\right) +C_F \ln\left(\frac{m_{Z'}^2}{m_t^2}\right)-C_F +C_A \ln\left(\frac{t_1}{u_1}\right) +C_A \ln\left(\frac{s m_{Z'}^2}{u_2^2}\right) \right. \nonumber \\ && \left. {}-(C_F+C_A)\ln\left(\frac{\mu_F^2}{m_{Z'}^2}\right) -\frac{\beta_0}{6}\right] \left[\frac{\ln^2(s_4/m_{Z'}^2)}{s_4}\right]_+ \nonumber \\ && \hspace{-25mm} {}+2(C_F+C_A)\left[\left(3C_F \ln\left(\frac{-t_2}{m_{Z'}^2}\right) -2C_F \ln\left(\frac{-u_1}{m_{Z'}^2}\right) -C_F \ln\left(\frac{m_{Z'}^2}{m_t^2}\right)+\frac{C_F}{4} +3C_A \ln\left(\frac{-u_2}{m_{Z'}^2}\right) \right. \right. \nonumber \\ && \left. {}+C_A \ln\left(\frac{u_1 m_{Z'}^2}{t_1 s}\right)-\frac{\beta_0}{4}\right) \ln\left(\frac{\mu_F^2}{m_{Z'}^2}\right) +\frac{\beta_0}{2} \ln\left(\frac{\mu_R^2}{m_{Z'}^2}\right) +\frac{1}{2}(C_F+C_A) \ln^2\left(\frac{\mu_F^2}{m_{Z'}^2}\right) \nonumber \\ && \left. -2(C_F+C_A) \zeta_2 \right] \left[\frac{\ln(s_4/m_{Z'}^2)}{s_4}\right]_+ \nonumber \\ && \hspace{-25mm} {}+(C_F+C_A)\left[\left(\frac{3\beta_0}{8}+\frac{3}{4}C_F -C_F\ln\left(\frac{-t_2}{m_{Z'}^2}\right) -C_A\ln\left(\frac{-u_2}{m_{Z'}^2}\right)\right)\ln^2\left(\frac{\mu_F^2}{m_{Z'}^2} \right) -\frac{\beta_0}{2}\ln\left(\frac{\mu_F^2}{m_{Z'}^2}\right)\ln\left(\frac{\mu_R^2}{m_{Z'}^2}\right)\right. \nonumber \\ && {}-2 \zeta_2 \left(2 C_F \ln\left(\frac{u_1}{t_2}\right) +C_F \ln\left(\frac{m_{Z'}^2}{m_t^2}\right)-C_F +C_A \ln\left(\frac{t_1}{u_1}\right)+C_A \ln\left(\frac{s m_{Z'}^2}{u_2^2}\right) \right. \nonumber \\ && \left. \left. \left. {}-(C_F+C_A)\ln\left(\frac{\mu_F^2}{m_{Z'}^2}\right)\right) +4(C_F+C_A)\zeta_3 \right] \left[\frac{1}{s_4}\right]_+ \right\} \, . \nonumber \\ \label{NNLOgqtZp} \end{eqnarray} The leading logarithms in the NNLO expansion are the $[\ln^3(s_4/m_{Z'}^2)/s_4]_+$ terms while the NLL are the $[\ln^2(s_4/m_{Z'}^2)/s_4]_+$ terms. Moreover, at NLL we determine in Eq. (\ref{NNLOgqtZp}) additional terms involving the scale. The cross section with the inclusion of the soft-gluon corrections through NNLO is denoted as approximate NNLO (aNNLO). \setcounter{equation}{0}\section{\label{Pheno} Phenomenological analysis} In the following sections we present the results of our phenomenological analysis in which we investigate the impact of the QCD corrections due to soft gluon emissions to the production of a single top quark in association with a $Z'$ for the case studies previously discussed. According to recent LHC Run II exclusion limits~\cite{Sirunyan:2018exx,Aad:2019fac}, extra neutral currents with masses $m_{Z'}\lesssim$ 4 TeV are disfavoured. In our analysis we consider final-state $Z'$s with masses ranging from 1 to 8 TeV where lighter $Z'$ masses are still included, because we wish to illustrate the behavior of the cross section and its scaling with the different phase-space suppression due to a final state with $Z'$ masses from low to high. \subsection{Comparison with existing results at NLO} We first illustrate a comparison of our aNLO calculation against other existing results at NLO. Then we discuss the matching of our aNNLO calculation to the exact NLO at fixed order in QCD. To validate the formalism at aNLO, we use $tZ$ production at the LHC in the presence of FCNC and compare the total cross section and scale dependence for the $gu \rightarrow tZ$ channel at NLO to the results of Ref.~\cite{Li:2011ek}. The comparison is summarized in Table~\ref{Comparison-Table} and was already documented in Ref.~\cite{Kidonakis:2017mfy}. \begin{table}[] \begin{center} \begin{tabular}{l l l l l} \hline \hline $\sigma_0^\textrm{aNLO}$ & $\sigma_0^\textrm{NLO}$ (Ref.~\cite{Li:2011ek}) & $(\sigma(\mu)/\sigma_0)_{ \textrm{aNLO}}$ & $(\sigma(\mu)/\sigma_0)_{ \textrm{NLO}}$ (Ref.~\cite{Li:2011ek})& \\ \hline 22.55 pb & 22.5 pb & ${}^{0.912 ~~ \mu=2 (m_Z + m_t)}_{1.103 ~~ \mu = (m_Z + m_t)/2}$ & ${}^{0.913~~ \mu=2 (m_Z + m_t)}_{1.112~~ \mu = (m_Z + m_t)/2} $ & \\ \hline \end{tabular} \caption{Total rate comparison for $gu \rightarrow tZ$ at the LHC 14 TeV: aNLO vs NLO from Ref.~\cite{Li:2011ek}. The cross section $\sigma_0$ is the default central value obtained using the central scale choice $\mu=m_Z + m_t$, i.e.~$\sigma_0 = \sigma(\mu = m_Z + m_t)$. The scale dependence is obtained by varying $\mu$ up and down by a factor of 2, i.e. $(m_Z + m_t)/2 \leq \mu \leq 2 (m_Z + m_t)$. CTEQ6M NLO PDFs~\cite{Pumplin:2002vw} are used. } \label{Comparison-Table} \end{center} \end{table} These numbers are in very good agreement (within 2 per mille) with Ref.~\cite{Li:2011ek} and can be checked in Table I and Fig. 6 respectively in that paper. They show that the soft-gluon approximation is excellent for these processes. As also noted in Ref.~\cite{Kidonakis:2017mfy}, the agreement between aNLO and NLO is also very good for the $gc \rightarrow tZ$ channel. A second independent cross check for the $gu \rightarrow tZ$ channel was made by using \textsc{MadGraph5\_aMC@NLO}~\cite{Alwall:2014hca} which provides both the total rate and the top-quark $p_T$ distribution. We have used the FCNC Madgraph module described in Refs.\cite{Degrande:2014tta,Durieux:2014xla} which employs a general approach to top-quark FCNC based on effective field theory. We fixed the parameters such that we could compare the cross section relative to the tensor interaction term only in the Lagrangian. We obtained the results illustrated in Fig.~\ref{aNLOvsNLO-top-pT} where the aNLO prediction is in very good agreement with the NLO calculation. \begin{figure}[htb!] \begin{center} \includegraphics[width=10.0cm]{aNLO-NLO-pTt.eps} \caption{aNLO vs NLO Madgraph top-quark $p_T$ distribution at the 14 TeV LHC. } \label{aNLOvsNLO-top-pT} \end{center} \end{figure} In the case of $tZ'$ production, our aNLO results have also been compared to the full NLO calculation at 7 TeV LHC energy provided in Ref.~\cite{Adelman:2012py}. In particular, we compared $K$-factors. It is important to notice that the Lagrangian used to obtain the results in Ref.~\cite{Adelman:2012py} only includes vector interaction contributions, e.g., ${\cal L}_{Z'}=(Q_{tU}/\sqrt{2}) \bar{U} \gamma^{\mu} g_R P_R t Z_{\mu}^\prime + ~h.c.$, where $Q_{tU}$ is a coupling factor, $g_R$ is a coupling constant, $P_R$ is the right-handed chiral projector, and $U$ denote the generic up-type quark. In this study, we consider only tensor interactions (cf. Eq.~(\ref{Langrangian})). Moreover, the authors of Ref.~\cite{Adelman:2012py} have used different PDFs, MSTW2008 \cite{Martin:2009iq}, and a different choice of central scale, $(m_{Z'}+m_t)/2$, than our choice of central scale, $m_{Z'}$. Therefore, to make a valid comparison between $K$-factors from vector and tensor interactions, we have adopted their PDFs and scale choices to make a comparison at 7 TeV. Because they use Run 1 LHC energies, the authors of Ref.~\cite{Adelman:2012py} only show results up to $m_{Z'}$ masses of 2000 GeV. In Fig.~\ref{K-fact-7tev} we display NLO/LO $K$-factors relative to vector interactions and aNLO/LO $K$-factors relative to tensor interactions for $\mu=(m_{Z'}+m_t)/2$, and a variation of that scale by a factor of two up and down for the NLO and aNLO corrections at these scales relative to the central LO result at $\mu=(m_{Z'}+m_t)/2$. We find that tensor interactions give $K$-factors at aNLO which are very similar in magnitude to those obtained by using vector interactions, but the scale dependence for the tensorial case is found to be somewhat smaller than (but consistent with) the vector case. We stress, however, that we do not expect exact agreement between the two cases due to the different Lagrangians involved. In the inset plot of Fig.~\ref{K-fact-7tev} we also display the additional enhancements from the aNNLO corrections, where in the numerator of aNNLO/aNLO we use NNLO PDFs and in the denominator we use NLO PDFs. In conclusion, we have shown that for $tZ$ production the soft-gluon corrections account for the overwhelming majority of the complete corrections and that the aNLO calculation is very trustworthy. This was already demonstrated for $tZ$ production in Ref.~\cite{Kidonakis:2017mfy} and it is also consistent with the fact that the NLO soft-gluon corrections approximate very well the complete NLO corrections for $t\gamma$~\cite{Forslund:2018qcp} production via anomalous couplings, as well as for top-pair~\cite{NKttbar} and single-top~\cite{NKsingletop} production. \subsection{Matching to the NLO theory at fixed order in QCD} The formalism utilized in this study is expected to work equally well in the case of $tZ'$ production, because it is essentially the same, the only difference being that the mass of the $Z'$ can have different values. Indeed, after performing the aNLO and NLO calculations for $tZ'$ production for a variety of collider energies and $Z'$ masses, we observed that the aNLO and the exact NLO results differ by a few percent. As expected, at large collider energies and large $m_{Z'}$ values, soft-gluon corrections account for the overwhelming majority of the QCD corrections, and the difference between the approximate and the exact NLO predictions is found to be very small. To further improve our theoretical predictions, we match our aNNLO prediction to the exact NLO theory at fixed order in QCD, and in the rest of this paper we show phenomenological results at NLO and aNNLO. The NLO fixed order theory prediction for both the FCNC and the stringy inspired $tZ'$ production is obtained with \textsc{MadGraph5\_aMC@NLO-v2.7.2}, which we have used to calculate both the total rate and the top-quark $p_T$ distributions. The approximate aNNLO theory prediction is obtained by matching to the NLO as follows: \begin{equation} \sigma_{aNNLO} = \left[\hat{\sigma}_{LO} + \hat{\sigma}_{NLO} + \hat{\sigma}_{aNNLO} \right]_{ij} \otimes f_i^{NNLO} \otimes f_j^{NNLO}\,, \end{equation} where the soft-gluon contributions from the aNNLO hard scattering are added on top of the fixed-order NLO. The matching procedure ensures a better control of kinematic regions of the phase space where soft-gluons are less dominant. \begin{figure}[htb!] \begin{center} \includegraphics[width=10.0cm]{Kfact-nlo-lo-MZpscan-gu-lhc7.eps} \caption{NLO $K$-factors for top FCNC with vector interactions and the aNLO $K$-factors for top FCNC with tensor interactions for the $gu\rightarrow tZ'$ channel at the LHC at 7 TeV. Scale variation refers to $(m_{Z'}+m_t)/4\leq \mu \leq m_{Z'}+m_t$. The inset plot also shows the aNNLO $K$-factors.} \label{K-fact-7tev} \end{center} \end{figure} \subsection{FCNC $Z'$s: $gu \rightarrow tZ'$ and $gc \rightarrow tZ'$} We first study $tZ'$ production via FCNC interactions with anomalous couplings. The partonic processes involved are $gu \rightarrow tZ'$ and $gc \rightarrow tZ'$, where the $Z'$ anomalously couples to the top quark and the $u$ and $c$ quarks through the flavor-changing coefficients $k_{tuZ'}/\Lambda$ and $k_{tcZ'}/\Lambda$, respectively. The scale $\Lambda$ is set equal to ten times the top quark mass $m_t$ and the couplings $k_{tuZ'}$ and $k_{tcZ'}$ are considered as parameters of the theory. As a case study we select $k_{tuZ'}=k_{tcZ'}=0.1$. Thus, in our results below we set $k_{tuZ'}/\Lambda=k_{tcZ'}/\Lambda=0.01/m_t$. We also set $m_t=172.5$ GeV. Recent experimental searches for and phenomenological studies of FCNC interactions between the top quark and a $Z$ boson can be found in Refs.~\cite{CMS:2016bss,Aad:2015uza,Khachatryan:2015att,Khanpour:2014xla,Hou:2017ozb}. We explore cross sections at 13 and 14 TeV LHC energies for a large range of $Z'$ masses, and also explore the cross sections as functions of $pp$ collider energy for future colliders. The theory predictions in this case are obtained by using the CT14 PDFs~\cite{Dulat:2015mca} which lead to the numerical results illustrated in Figs.~\ref{gutZp13}-\ref{K-fact-scaleunc}. In this case, PDF induced uncertainties are calculated at the 68\% confidence level (C.L.) (see Appendix \ref{AppA} for a discussion on PDF uncertainties). The initial-state parton combinations $g(x_1)u(x_2) + g(x_2)u(x_1)$ and $g(x_1)c(x_2) + g(x_2)c(x_1)$ are probed in various kinematic regions depending on the collider center-of-mass energy and on the mass of the $Z'$. At $\sqrt{S}= 13$ TeV and $1\lesssim M_{Z'}\lesssim 8$ TeV, one probes large $x$ values $x\geq 0.1$ where the current PDFs are not well constrained and their uncertainties are large. At higher collider energies $\sqrt{S}= 100$ TeV, one probes $10^{-4} \lesssim x \lesssim 0.1$ for $ M_{Z'}\approx 1$ TeV, and $0.01 \lesssim x \lesssim 0.1$ for $M_{Z'}\approx 8$ TeV. \begin{figure}[th!] \begin{center} \includegraphics[width=8.3cm]{gutZp13lhcnewplot.eps} \hspace{1mm} \includegraphics[width=8.3cm]{gctZp13lhcnewplot.eps} \caption{Total cross sections at 13 TeV LHC energy for (left) $gu\rightarrow tZ'$ with anomalous $t$-$u$-$Z'$ coupling and (right) $gc\rightarrow tZ'$ with anomalous $t$-$c$-$Z'$ coupling. The inset plots display $K$-factors. Here CT14NNLO PDFs are used for the LO, NLO, and aNNLO calculations to show the enhancement due to hard-scattering contributions.} \label{gutZp13} \end{center} \end{figure} The total cross sections at collider energies of 13 TeV are illustrated in Fig.~\ref{gutZp13} where we show the theory predictions at LO, NLO, and aNNLO for the process $gu\rightarrow tZ'$ with anomalous $k_{tuZ'}$ coupling, and the process $gc\rightarrow tZ'$ with anomalous $k_{tcZ'}$ coupling, as functions of $Z'$ mass. Here CT14NNLO PDFs are used for the LO, NLO, and aNNLO calculations to show soft-gluon enhancements in the hard-scattering contributions with respect to the Born cross section. The factorization and renormalization scales are equal and set to $\mu=m_{Z'}$. We observe a very strong dependence of the cross section on the $Z'$ mass. The cross section drops over many orders of magnitude as the $Z'$ mass varies from 1 TeV to 6 TeV. The cross section for $gc \rightarrow tZ'$ is significantly smaller than for $gu\rightarrow tZ'$. The inset plots show the NLO/LO and aNNLO/LO $K$-factors with scale uncertainty bands which are obtained by varying $\mu$ in the interval $[1/2\mu , 2\mu]$ in the numerator. The $K$-factors are large and increase with larger $Z'$ masses, as expected. The NLO corrections are large and furthermore the additional aNNLO corrections are very significant. We also provide numerical values for the $gu\rightarrow tZ'$ cross section and $K$-factors in Table \ref{tZ'table-gu} of Appendix \ref{AppC}. \begin{figure}[th!] \begin{center} \includegraphics[width=8.3cm]{gutZp14lhcnewplot.eps} \hspace{1mm} \includegraphics[width=8.3cm]{gctZp14lhcnewplot.eps} \caption{Total cross sections at 14 TeV LHC energy for the (left) $gu\rightarrow tZ'$ and (right) $gc\rightarrow tZ'$ processes with anomalous couplings. The inset plots display $K$-factors. Here CT14NNLO PDFs are used for the LO, NLO, and aNNLO calculations to show the enhancement due to hard-scattering contributions.} \label{gutZp14} \end{center} \end{figure} The corresponding results at 14 TeV energy are shown in Fig.~\ref{gutZp14}. The cross sections are of course larger than at 13 TeV, but the dependence on the $Z'$ mass and the size of the corrections are very similar. \begin{figure} \begin{center} \includegraphics[width=8.3cm]{MZpscan-multiple-S-gu-nnlopdfs-vs-nlopdfs-multi-new.eps} \hspace{1mm} \includegraphics[width=8.3cm]{MZpscan-multiple-S-gc-nnlopdfs-vs-nlopdfs-multi-new.eps} \caption{Total cross sections for the (left) $gu \rightarrow tZ'$ and (right) $gc\rightarrow tZ'$ processes with anomalous couplings. The plots show results including CT14 PDF uncertainties for several center-of-mass energies of the $pp$ collision as a function of $Z'$ mass. The aNNLO cross section is obtained with CT14NNLO PDFs while the NLO with CT14NLO. The CT14 PDF uncertainties are at the 68\% C.L.. The inset plots show the $\sigma_{aNNLO}/\sigma_{NLO}$ $K$-factors.} \label{gutZpscan} \end{center} \end{figure} In Fig.~\ref{gutZpscan} we show the total cross sections at NLO and aNNLO for the processes $gu\rightarrow tZ'$ and $gc\rightarrow tZ'$ at 13, 27, 50, and 100 TeV collider energies together with CT14 PDF uncertainties evaluated at the 68\% confidence level (C.L.). In this case, the aNNLO total cross sections are obtained with CT14NNLO PDFs, while the NLO's are obtained with CT14NLO. The inset plots show the $\sigma_{aNNLO}/\sigma_{NLO}$ $K$-factors. We note that the $\sigma_{aNNLO}/\sigma_{LO}$ $K$-factors are not shown here because there are no CT14 PDFs at LO. We observe that the $\sigma_{aNNLO}/\sigma_{NLO}$ $K$-factors provide large corrections for large values of $m_{Z'}$, and the corrections decrease as the collider energy increases. The induced PDF uncertainty of both $gu$ and $gc$ channels is larger at lower collider energy and high $m_{Z'}$ where PDFs are weakly constrained. \begin{figure}[th!] \begin{center} \includegraphics[width=8.3cm]{MZpscan-multiple-S-gu-multi-new.eps} \hspace{1mm} \includegraphics[width=8.3cm]{MZpscan-multiple-S-gc-multi-new.eps} \caption{Total cross sections for the $gu \rightarrow tZ'$ (left) and the $gc \rightarrow tZ'$ (right) processes with anomalous couplings. The plots show results with CT14 NNLO PDF uncertainties at 68\% C.L. for several center-of-mass energies of the $pp$ collision as a function of $m_{Z'}$.} \label{MZpscan-ct14nnlo} \end{center} \end{figure} Figure~\ref{MZpscan-ct14nnlo} shows total cross section predictions at 13, 27, 50, and 100 TeV collider energies using CT14NNLO PDFs at all orders for FCNC $tZ'$ production to show the enhancement due to soft gluons in the perturbative series. \begin{figure} \begin{center} \includegraphics[width=8.3cm]{gutZprootSnewplot.eps} \hspace{1mm} \includegraphics[width=8.3cm]{gctZprootSnewplot.eps} \caption{Total cross sections for the (left) $gu\rightarrow tZ'$ and (right) $gc\rightarrow tZ'$ processes with anomalous couplings. The plots show results as a function of collider energy for three choices of $Z'$ mass, 3, 5, and 8 TeV. The inset plots display $K$-factors. CT14NNLO PDFs are used.} \label{gutZprootS} \end{center} \end{figure} The behavior of the cross section with collider energy is illustrated in Fig.~\ref{gutZprootS}, where we show results at LO, NLO, and aNNLO for the $gu$ and $gc$ channels as functions of the collider energy up to 100 TeV for three choices of $Z'$ mass, $m_{Z'}=3$, 5, and 8 TeV. Here, the LO, NLO, and aNNLO cross sections are obtained with CT14NNLO PDFs to show enhancement in the hard scattering due to soft gluon corrections. The cross sections are smaller for larger $Z'$ masses due to phase-space suppression. The inset plots show the NLO/LO and aNNLO/LO $K$-factors. As expected, the $K$-factors are larger at smaller energies and also for higher $Z'$ masses, since we are then closer to threshold. \begin{figure} \begin{center} \includegraphics[width=8.4cm]{kukcscan-mzp-5TeV-lhc13-2D-new.eps} \includegraphics[width=8.4cm]{kukcscan-mzp-8TeV-lhc13-2D-new.eps} \\ \includegraphics[width=8.4cm]{kukcscan-mzp-5TeV-sqrtS100TeV-2D-new.eps} \includegraphics[width=8.4cm]{kukcscan-mzp-8TeV-sqrtS100TeV-2D-new.eps} \caption{Total cross sections for the $(gc+gu)\rightarrow tZ'$ process in a 2D contour plot. The insets show aNNLO results as a function of the anomalous couplings $k_{tuZ'}/\Lambda$ and $k_{tcZ'}/\Lambda$ (given in units of inverse $m_t$), in $pp$ collisions at $\sqrt{S}$=13 and 100 TeV. CT14NNLO PDFs are used.} \label{coupling-scan} \end{center} \end{figure} In the case of $tZ'$ production with FCNC couplings, the anomalous couplings entering both channels of the cross section are considered as free parameters. We have therefore performed a two dimensional scan to assess the sensitivity of the cross section. In Fig.~\ref{coupling-scan} we show a case study in which we plot aNNLO total cross sections as functions of the couplings $k_{tuZ'}/\Lambda$ and $k_{tcZ'}/\Lambda$, at a collider energies of 13 and 100 TeV, for different values of $m_{Z'}$. We notice that if we let both couplings to vary in $10^{-5} \leq k/\Lambda\leq 0.1$ TeV$^{-1}$, the cross section spans several orders of magnitude. The cross section suppression is larger for larger values of $m_{Z'}$. \subsubsection{Top-quark $p_T$ distributions for FCNC $Z'$s } It is interesting to study kinematic distributions such as the top-quark $p_T$ differential distribution, $d\sigma/dp_T$, and how $Z's$ of different masses affect the $p_T$ suppression in various kinematic ranges. We illustrate the top-quark $p_T$ distributions, calculated by a numerical integration of the double-differential distribution, in Fig.~\ref{ptgutZp100}. Results for the $gu \rightarrow tZ'$ and $gc \rightarrow tZ'$ processes at a collider energy of 100 TeV are shown at LO, NLO, and aNNLO, obtained with CT14NNLO PDFs, for three choices of the $Z'$ mass of 3, 5, and 8 TeV. \begin{figure} \begin{center} \includegraphics[width=8.3cm]{pttopgutZp100tevnewplot.eps} \hspace{1mm} \includegraphics[width=8.3cm]{pttopgctZp100tevnewplot.eps} \caption{Top-quark $p_T$ distributions for the (left) $gu \rightarrow tZ'$ and (right) $gc\rightarrow tZ'$ processes with anomalous couplings for $m_{Z'}=3$, 5, and 8 TeV at 100 TeV $pp$ collider energy. Inset plots: NLO/LO and aNNLO/LO K-factors. CT14NNLO PDFs are used.} \label{ptgutZp100} \end{center} \end{figure} The NLO corrections are large and furthermore the additional aNNLO corrections are important. The $p_T$ distributions decrease quickly as $m_{Z'}$ is increased, but they are non-negligible even for large $Z'$ masses, indicating that the number of events predicted by these models can be validated at the high-luminosity FCC or SppC colliders. The $K$-factors, shown in the inset plots, are significant and their value depends on $m_{Z'}$ and on the phase-space supression. \subsubsection{Cross section and PDF correlations} Next, we explore the extent of correlation between the PDFs and the aNNLO cross section for these processes in $pp$ collisions at $\sqrt{S}$=13 and 100 TeV. PDF correlations are important because they give us information about the kinematic region in which PDFs are probed and for example, they give us indication of the impact of the gluon at different values of the momentum fraction $x$. In order to set tighter constraints on $Z'$s models it is important to understand how PDF uncertainties come into play and how to improve their precision through dedicated QCD global analyses. In particular, in Fig.~\ref{corr-cos-sgu-sgc} we show the correlation cosine between the gluon (and the $u$ quark) and the total cross section for the $gu \rightarrow t Z'$ process as a function of the momentum fraction $x$ at the 68\% CL at $\sqrt{S}=$ 13 and 100 TeV. We have chosen the $gu$ channel as it provides the dominant contribution. The definition of the correlation cosine between two quantities determined within the Hessian method is given in Appendix~\ref{AppB}. At collider energies of 13 TeV, we observe a strong correlation ($\cos{\phi}\geq 0.8$) between the gluon and the $gu \rightarrow t Z'$ cross section at large $x\geq 0.1$ as expected, and the correlation peak shifts towards larger $x$ values for larger $m_{Z'}$. Anti-correlation of approximately 50\% in the $10^{-4} \leq x\leq 10^{-2}$ interval is also observed. The correlation between the $u$ quark and the cross section is much milder and less than 50\% at very large $x$. These patterns change as we move to higher collider energies, where for the gluon the correlation peak for each value of $m_{Z'}$ is shifted to lower $x$-values, while for the $u$ quark correlations are slightly more pronounced. Besides the correlation with PDFs, important information can also be gathered from the study of simultaneous uncertainty boundaries of the cross section of the $gu$ and $gc$ channels. The allowed regions are represented by correlation ellipses which can be compared to pseudo data in BSM simulations and explore the implications of the PDFs for this process. In Fig.~\ref{ellipse-sgu-sgc} and \ref{ellipse-sgu-sgc-100TeV} we show the elliptical confidence regions, at 68\% CL, in $pp$ collisions at 13 and 100 TeV, for $m_{Z'}=1$, 3, 5, and 8 TeV. These can be used to read off PDF uncertainties and correlations for each pair of cross sections. At $\sqrt{S}=$ 13 TeV, we notice that the two channels are highly correlated and the induced PDF uncertainties on the $\sigma_{gc}$ channel are very large for this choice of the collider energy. This is reflected by the fact that there is a small portion of the ellipse where the PDF induced errors on the cross sections are larger than the cross section central value itself, allowing for negative values. At $\sqrt{S}=$ 100 TeV, the $gu$ and $gc$ channels are still highly correlated, but the induced PDF uncertainties on both the cross sections are smaller as in this kinematic domain the PDFs are probed at intermediate $x$ where they are better constrained. Next, we study the impact of the scale and PDF uncertainties on the aNNLO/LO $K$-factors as functions of the collider energy for large $\sqrt{S}$ values and different values of $m_{Z'}$. In Fig.~\ref{K-fact-PDFunc} and \ref{K-fact-scaleunc}, we illustrate the $K$-factors for the $gu$ and $gc$ channels with CT14NNLO PDF and scale uncertainties respectively. Scale variation refers to $m_{Z'}/2\leq \mu \leq 2 m_{Z'}$ as before. In Fig.~\ref{K-fact-PDFunc} the PDF uncertainties for each $m_{Z'}$ value are shown using bands with different hatches and color. At collider energies below 20 TeV PDF uncertainties are large because PDFs are probed in the large-$x$ region. In the $gc$ channel, PDF uncertainties are dominant because the charm-quark PDF is less constrained with respect to the gluon and $u$-quark. In Fig.~\ref{K-fact-scaleunc} the scale dependence in the aNNLO $K$-factors for the $gu$ and $gc$ channels is illustrated separately. \begin{figure}[tbh!] \begin{center} \includegraphics[width=8.3cm]{cosphi-corr-gluon-Mzpscan-lhc13-new.eps} \hspace{1mm} \includegraphics[width=8.3cm]{cosphi-corr-up-Mzpscan-lhc13-new.eps} \\ \includegraphics[width=8.3cm]{cosphi-corr-gluon-Mzpscan-lhc100-new.eps} \hspace{1mm} \includegraphics[width=8.3cm]{cosphi-corr-up-Mzpscan-lhc100-new.eps} \caption{Correlation cosine at $\sqrt{S}$=13 and 100 TeV between $\sigma_{gu\rightarrow tZ'}$ and the gluon as a function of $x_{\textrm{gluon}}$ (left column) and $\sigma_{gu\rightarrow tZ'}$ and the up-quark as a function of $x_{\textrm{up}}$ (right column). The four panels show aNNLO results rescaled at the 68\% C.L. for different values of the $Z^\prime$ mass.} \label{corr-cos-sgu-sgc} \end{center} \end{figure} \begin{figure}[tbh!] \begin{center} \includegraphics[width=8.3cm]{ellipse-sgu-sgc-Mzp1TeV-LHC13-new.eps} \hspace{1mm} \includegraphics[width=8.3cm]{ellipse-sgu-sgc-Mzp3TeV-LHC13-new.eps}\\ \includegraphics[width=8.3cm]{ellipse-sgu-sgc-Mzp5TeV-LHC13-new.eps} \hspace{1mm} \includegraphics[width=8.3cm]{ellipse-sgu-sgc-Mzp8TeV-LHC13-new.eps} \caption{Correlation ellipses of the $gu\rightarrow tZ'$ and $gc\rightarrow tZ'$ channels at $\sqrt{S}=13$ TeV. The figures show CT14NNLO PDF induced uncertainty boundaries of the aNNLO results. Uncertainties are rescaled at the 68\% C.L. for different values of the $Z^\prime$ mass.} \label{ellipse-sgu-sgc} \end{center} \end{figure} \begin{figure}[tbh!] \begin{center} \includegraphics[width=8.3cm]{ellipse-sgu-sgc-Mzp1TeV-LHC100-new.eps} \hspace{1mm} \includegraphics[width=8.3cm]{ellipse-sgu-sgc-Mzp3TeV-LHC100-new.eps}\\ \includegraphics[width=8.3cm]{ellipse-sgu-sgc-Mzp5TeV-LHC100-new.eps} \hspace{1mm} \includegraphics[width=8.3cm]{ellipse-sgu-sgc-Mzp8TeV-LHC100-new.eps} \caption{Same as in Fig.~\ref{ellipse-sgu-sgc}, but for $pp$ collisions at $\sqrt{S}=100$ TeV.} \label{ellipse-sgu-sgc-100TeV} \end{center} \end{figure} \begin{figure}[tbh!] \begin{center} \includegraphics[width=8.2cm]{Kfact-nnlo-lo-SqrtS-MZpscan-gu-set2-sqrtS-new.eps} \includegraphics[width=8.2cm]{Kfact-nnlo-lo-SqrtS-MZpscan-gc-set2-sqrtS-new.eps} \caption{$K$-factors with CT14NNLO PDF uncertainties (68\% C.L.) for the $gu\rightarrow tZ'$ and $gc\rightarrow tZ'$ channels. The figures show a scan in the center of mass energy of the collisions $\sqrt{S}$ for different values of the $Z^\prime$ mass. } \label{K-fact-PDFunc} \end{center} \end{figure} \begin{figure}[th!] \begin{center} \includegraphics[width=8.3cm]{Kfact-nnlo-lo-SqrtS-MZpscan-gu-set2-scaleonly-new.eps} \includegraphics[width=8.3cm]{Kfact-nnlo-lo-SqrtS-MZpscan-gc-set2-scaleonly-new.eps} \caption{Scale uncertainty in the aNNLO $K$-factors for the $gu\rightarrow tZ'$ and $gc\rightarrow tZ'$ channels. The figures show a scan in the center of mass energy of the collisions $\sqrt{S}$ for different values of the $Z^\prime$ mass. Scale variation refers to $m_{Z'}/2\leq \mu \leq 2 m_{Z'}$. CT14NNLO PDFs are used.} \label{K-fact-scaleunc} \end{center} \end{figure} \clearpage \subsection{\label{Stringy-Zprimes} String-inspired $Z'$s: $gt \rightarrow tZ'$} In this section we discuss the phenomenological results obtained from the study of $tZ'$ production where the $Z'$ originates from a low-energy realization of string-inspired models. The interaction Lagrangian in Sec.~\ref{stringyZprime}, and the choice of the parameters we have examined, are based on the models published in Refs.~\cite{Faraggi:2015iaa,Coriano:2008wf}. These models have not been searched for by the ATLAS and CMS collaborations to the best of our knowledge, therefore the current limits on the $Z'$ mass and couplings should in principle not be applied here. The $Z'$ models described in Refs.~\cite{Faraggi:2015iaa,Coriano:2008wf} allow for non-sequential solutions (i.e. charge assignments which are not proportional to the hypercharge) that are phenomenologically interesting and could in principle be considered in future analyses by both ATLAS and CMS. An accurate determination of the $gt \rightarrow tZ'$ cross section can play an important role to set constraints on the couplings of $Z'$ to the fermion sector. In fact, this process can in principle be used together with $Z'$ production in Drell-Yan to remove the degeneracy between quark and lepton couplings~\cite{Petriello:2008zr,Petriello:2008pu}. The leading-order cross section is given by the $s$- and $t$-channels of the $gt \rightarrow tZ'$ process and the structure of the couplings is given in Sec.~\ref{stringyZprime}. The $gt \rightarrow tZ'$ process with $m_{Z'}$ in the TeV range requires the top-quark PDF in the initial state. In our phenomenological application, $\mu=m_{Z'}\gg m_t$ and we consider the top quark as an active flavor inside the proton with very good approximation. Therefore, in the rest of this analysis we work with the $6$-flavor scheme and use the NNPDF3.1 PDFs~\cite{Ball:2017nwa} with $n_f=6$ and $\alpha_s(m_Z)=0.118$, where $n_f$ is the number of active flavors. We set $m_t=0$ in the initial state lines in the calculation of the LO cross section. In this case, PDF uncertainties are calculated at 1-$\sigma$ C.L. (see Appendix~\ref{AppA}) which is almost identical to the 68\% C.L. in absence of statistical fluctuations in the determination of the PDFs. In Fig.~\ref{top-quark-PDF} we illustrate the top-quark PDF uncertainty as a function of $x$ for different values of the final-state $Z'$ mass. The $gt \rightarrow tZ'$ process probes the top-quark and gluon PDFs at large $x$ where uncertainties are large at the LHC Run II collision energies. Precision measurements in the extended kinematic domain of the future FCC-eh collider will allow us to extract PDFs at large $x$ for the individual quark flavors at the percent level precision. The precision of the top-quark PDF will be improved in this kinematic region enhancing the FCC-hh discovery potential of $Z'$s with mass of ${\cal O}$(10) TeV also in rare processes. \begin{figure}[htb!] \begin{center} \includegraphics[width=8cm]{./xtop-ratio-PDF.eps} \includegraphics[width=8cm]{./xtop-PDF.eps} \caption{The error bands represent NNPDF3.1 NNLO $n_f=6$ PDF uncertainties evaluated at the 1-$\sigma$ C.L.. } \label{top-quark-PDF} \end{center} \end{figure} The left plot of Fig.~\ref{tZp-mazscan-nnlopdfs-vs-nlopdfs} shows the NLO and aNNLO total cross section for the $gt\rightarrow tZ'$ process as a function of $m_{Z'}$ at collider energies $\sqrt{S} = 13, 27, 50, 100$ TeV. The error bands represent the induced PDF uncertainties on the cross section at 1-$\sigma$ C.L. obtained by using NNPDF3.1 $n_f = 6$ PDFs. The aNNLO prediction is obtained using NNPDF3.1 NNLO $n_f=6$ PDFs, while the NLO is obtained using NNPDF3.1 NLO $n_f=6$ PDFs. The LO cross section is not shown here because the NNPDF3.1 $n_f=6$ PDFs at LO are not available. The inset plot shows the $\sigma_{aNNLO}/\sigma_{NLO}$ $K$-factors from where we observe that the $K$-factors are large and they increase as $m_{Z'}$ increases, and they decrease when the collider energy increases, as for the case of the FCNC $Z'$s. \begin{figure} \begin{center} \includegraphics[width=8cm]{MAZpscan-nnlopdfs-vs-nlopdfs-multi-new.eps} \includegraphics[width=8cm]{MAZpscan-multi-new.eps} \caption{(Left) Total cross sections for the $gt\rightarrow tZ'$ process at NLO and aNNLO as a function of the mass of the $Z'$ for various collider energies. The aNNLO result is obtained using NNPDF3.1 NNLO $n_f=6$ PDFs, while NLO is obtained using the same PDFs at NLO. (Right) Total cross sections for the $gt\rightarrow tZ'$ process at LO, NLO, and aNNLO where all results use NNPDF3.1 NNLO PDFs. In both plots the error bands represent PDF uncertainties at 1-$\sigma$ C.L., and the inset plots show $\sigma_{aNNLO}/\sigma_{NLO}$ $K$-factors.} \label{tZp-mazscan-nnlopdfs-vs-nlopdfs} \end{center} \end{figure} In the plot on the right of Fig.~\ref{tZp-mazscan-nnlopdfs-vs-nlopdfs}, the cross section is obtained by convoluting hard scatterings at LO, NLO, and aNNLO, with NNLO PDFs in order to show the enhancement due to the hard-scattering ontributions only. The $Z'$ coupling $g_{Z'}$ is considered as a free parameter and as a case study we choose $g_{Z'}=1$ as the default choice. A $g_{Z'}$ parameter scan is illustrated in Fig.~\ref{gz-scan-tZp-mazscan} (left) where the aNNLO cross section is plotted as a function of $\sqrt{S}$ for different values of $m_{Z'}$ which correspond to bands with different dashing. We explore $g_{Z'}$ variations in $0.01\leq g_{Z'}\leq 1.5$ and observe that when $g_{Z'}$ varies the cross section is basically rescaled and it spans approximately two orders of magnitude. Moreover, for comparison purposes, we consider the production of a sequential $Z'$ as a commonly-used point of reference. In Fig.~\ref{gz-scan-tZp-mazscan} (right) we illustrate a comparison between aNNLO total cross sections for the production of string-inspired $Z'$s and the production of sequential $Z'$s, for different values of the collider energy. The sequential $Z'$s are extra neutral vector bosons which have vector and axial-vector couplings equal to those of the SM $Z$-boson, but such that their right-handed and left-handed couplings to quarks are defined up to a constant factor which we set equal to $g_{Z'}$, e.g., $g^{(Z')}_{R,L} = g_{Z'}(g_V\pm g_A)$. In this specific comparison we consider $Z'$ masses larger than 4 TeV because sequential $Z'$s are currently excluded for smaller masses~\cite{Aaboud:2017buh,CMS:2016abv}. As expected, the shapes in the two models are identical. \begin{figure} \begin{center} \includegraphics[width=8.2cm]{gzp-SqrtS-MAZpscan-new1.eps} \includegraphics[width=8.2cm]{MseqZpscan-vs-Mazp-scan-new.eps} \caption{Left: Scan of the $g_{Z'}$ parameter for the $gt\rightarrow tZ'$ process. The plot shows results of the total cross section at aNNLO as a function of collider energy. Bands with different dashing represent $Z'$ mass values of 3, 5, and 8 TeV. Right: Comparison between string-inspired $Z'$s and sequential $Z'$s for different values of the collider energy at aNNLO.} \label{gz-scan-tZp-mazscan} \end{center} \end{figure} Prospects at the LHC at 13 and 14 TeV collision energies are shown in Fig.~\ref{tZp13-14-Kfac} where the inset plots show the NLO/LO and aNNLO/LO $K$-factors. Here, LO, NLO, and aNNLO cross sections are all obtained by using NNPDF3.1 NNLO $n_f=6$ PDFs to show the soft-gluon enhancement in the hard scattering. We note the large effect of the higher-order corrections, which more than triple the LO result for a 6 TeV $Z'$ mass. We also provide numerical values for the $gt\rightarrow tZ'$ cross section and $K$-factors at 13 TeV energy in Table \ref{tZ'table-gt} of Appendix~\ref{AppC}. \begin{figure} \begin{center} \includegraphics[width=8cm]{tZp13lhcnewplot.eps} \hspace{2mm} \includegraphics[width=8cm]{tZp14lhcnewplot.eps} \caption{Total cross section for the $gt\rightarrow tZ'$ process at the LHC 13 TeV (left) and 14 TeV (right). LO, NLO and NNLO calculations are obtained using NNPDF3.1 NNLO $n_f=6$ PDFs.} \label{tZp13-14-Kfac} \end{center} \end{figure} Total cross section results as functions of the collider energy up to 100 TeV for different values of $m_{Z'}$ are given in Fig.~\ref{tZprootS}. The inset plot shows the NLO/LO and aNNLO/LO $K$-factors where NNPDF3.1 NNLO $n_f=6$ PDFs are used for LO, NLO, and aNNLO calculations. While the cross sections get smaller with increasing $Z'$ mass, the $K$-factors get larger because this kinematic region is closer to the partonic threshold. \begin{figure} \begin{center} \includegraphics[width=10cm]{tZprootSnewplot.eps} \caption{Total cross sections for the $gt\rightarrow tZ'$ process. The plot shows results as a function of collider energy for three choices of $Z'$ mass, 3, 5, and 8 TeV. The inset plot displays $K$-factors. NNPDF3.1 NNLO $n_f=6$ PDFs are used for LO, NLO, and aNNLO calculations. Cross sections get smaller with increasing $Z'$ mass while $K$-factors get larger.} \label{tZprootS} \end{center} \end{figure} In the left plot of Fig.~\ref{tZprootS-PDFunc-scaleunc} we illustrate the induced NNPDF3.1 $n_f=6$ NNLO PDF uncertainty on the $\sigma_{\textrm{aNNLO}}$ total cross section which we normalize to $\sigma_{\textrm{LO}}$ to obtain $K$-factors. Here the LO cross section is also obtained with NNLO PDFs. The large uncertainty of the top-quark NNLO PDF dominates at all collider energies and for every value of $m_{Z'}$. In the plot on the right of Fig.~\ref{tZprootS-PDFunc-scaleunc} we show the scale uncertainty due to factorization scale variation in $m_{Z'}/2\leq \mu \leq 2 m_{Z'}$. As mentioned in previous sections, the $K$-factors here are defined as $\sigma_{\textrm{aNNLO}}(\mu)/\sigma_{\textrm{LO}}$ where $\sigma_{\textrm{LO}}$ is obtained using the default central choice $\mu=m_{Z'}$ and NNPDF3.1 $n_f=6$ NNLO PDFs. \begin{figure} \begin{center} \includegraphics[width=8.3cm]{Kfact-nnlo-lo-SqrtS-MAZpscan-new.eps} \includegraphics[width=8.3cm]{Kfact-nnlo-lo-SqrtS-MAZpscan-set2-scaleonly-new.eps} \caption{Left: $K$-factors for the $gt\rightarrow tZ'$ process with NNPDF3.1 $n_f=6$ PDF uncertainties. The plot shows aNNLO/LO results as a function of $\sqrt{S}$ for three choices of $Z'$ mass, 3, 5, and 8 TeV. Right: $K$-factors with scale uncertainty bands. Scale variation refers to $m_{Z'}/2\leq \mu \leq 2 m_{Z'}$. } \label{tZprootS-PDFunc-scaleunc} \end{center} \end{figure} \subsubsection{Top-quark $p_T$ distributions for string-inspired $Z'$s} In this section we show the top-quark $p_T$ distributions for this process. Fig.~\ref{pttZp100} shows the top-quark $p_T$ distributions in the $gt \rightarrow tZ'$ process at LO, NLO, and aNNLO for different $m_{Z'}$ values at a collider energy of 100 TeV. NNPDF3.1 NNLO $n_f=6$ PDFs are used for LO, NLO, and aNNLO calculations to emphasize the enhancement in the hard scattering contribution. The $K$-factors are shown in the inset plot. \begin{figure} \begin{center} \includegraphics[width=10cm]{pttoptZp100tevnewplot.eps} \caption{Top-quark $p_T$ distributions for the $gt\rightarrow tZ'$ process at 100 TeV $pp$ collider energy for $m_{Z'}=3$, 5, and 8 TeV. NNPDF3.1 NNLO $n_f=6$ PDFs are used for LO, NLO, and aNNLO calculations.} \label{pttZp100} \end{center} \end{figure} \clearpage \setcounter{equation}{0}\section{\label{Conclusions} Conclusions} We have studied $tZ'$ production in various BSM models at hadron colliders. We performed a phenomenological QCD analysis where we scrutinized $tZ'$ production in the presence of FCNC and in the case in which the extra $Z'$ is generated within a low-energy realization of string theory models. We have calculated theoretical predictions for cross sections and top-quark $p_T$ distributions that include higher-order soft-gluon corrections. In particular, theory predictions are obtained at aNNLO in QCD by extending the soft-gluon resummation formalism to the case in which a top quark is produced in association with a heavy neutral vector boson in $pp$ collisions at energies that relevant for the LHC and for future new-generation hadron colliders like FCC-hh and SppC. We have found that QCD corrections due to soft-gluon emissions are considerable and need to be included in precision studies. We have investigated the impact of uncertainties due to proton PDFs as well as uncertainties due to scale variation. PDFs uncertainties represent the major source of uncertainty in this analysis. Moreover, we explored the parameter space for the BSM models we scrutinized by performing parameter scans and studying the sensitivity of the cross section to parameter changes. We have found that the total $tZ'$ cross section has large sensitivity on the mass of the $Z'$. These theoretical results will be useful for $tZ'$ production searches at the LHC and future hadron colliders. \setcounter{equation}{0}\section*{Acknowledgements} We thank Gauthier Durieux and Fabio Maltoni for correspondence and suggestions about the use of Madgraph5. The work of M.G. is supported by the National Science Foundation under Grant No. PHY 1820818. The work of N.K. is supported by the National Science Foundation under Grant No. PHY 1820795. Abstract: We study the production of a single top quark in association with a heavy extra $Z'$ at hadron colliders in new physics models with and without flavor-changing neutral-current (FCNC) couplings. We use QCD soft-gluon resummation and threshold expansions to calculate higher-order corrections for the total cross section and transverse-momentum distributions for $t Z'$ production. The impact of the uncertainties due to the structure of the proton and scale dependence is also analyzed.

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