geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

symmetric determinantal representation of polynomials abstract we give an elementary proof only using linear algebra of a result due to helton mccullough and vinnikov which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil we give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2

bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

asymptotically simple solutions of the vacuum einstein equations in even dimensions we show that a set of conformally invariant equations derived from the fefferman graham tensor can be used to construct global solutions of vacuum einstein equations in all even dimensions this gives in particular a new simple proof of friedrich s result on the future hyperboloidal stability of minkowski space time and extends its validity to even dimensions

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

inverse numerical range and determinantal representation abstract the numerical range w a of an n n matrix a is the collection of quadratic forms z a over unit sphere 1 the inverse numerical range problem aims to find a unit vector which corresponds to a given point z of the numerical range w a given a matrix a the helton vinnikov theorem produces a symmetric matrix s so that a and s have the same numerical range w a w s in this paper we investigate the inverse numerical range problem for the boundary points and points on the boundary generating curve of the numerical range in place of the construction of unit vectors satisfying z a we express the kernel vector function of the linear pencil x s y s z i n as a function on the abel jacobi variety of the associated elliptic curve of a the kernel function plays a key role for the inverse numerical problem we perform this process when s is a generic 3 3 symmetric matrix

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

real zero polynomials and a horn s problem abstract a horn s problem concerns finding two self adjoint matrices so that a b and a b have prescribed spectrum in this paper we show how it connects to an interpolation problem for two variable real zero polynomials and a tracial moment problem in addition we outline an algorithm to construct a pair a b

propagation de l analyticite des solutions de systemes hyperboliques non lineaires on montre que si l on se place dans un domaine ou la solution est a priori assez reguliere la regularite analytique s etend des donnees de cauchy a la solution selon la geometrie des domaines d influence de l equation comme dans le cas lineaire en particulier si les donnees de cauchy sont analytiques une solution est analytique jusqu au premier choc

first order conditions for semidefinite representations of convex sets defined by rational or singular polynomials a set is called semidefinite representable or semidefinite programming sdp representable if it equals the projection of a higher dimensional set which is defined by some linear matrix inequality lmi this paper discusses the semidefinite representability conditions for convex sets of the form s mathcal d f x in mathcal d f x geq 0 here mathcal d x in mathbb r n g 1 x geq 0 ldots g m x geq 0 is a convex domain defined by some nice concave polynomials g i x they satisfy certain concavity certificates and f x is a polynomial or rational function when f x is concave over mathcal d we prove that s mathcal d f has some explicit semidefinite representations under certain conditions called preordering concavity or q module concavity which are based on the positivstellensatz certificates for the first order concavity criteria f u nabla f u t x u f x geq 0 quad forall x u in mathcal d when f x is a polynomial or rational function having singularities on the boundary of s mathcal d f a perspective transformation is introduced to find some explicit semidefinite representations for s mathcal d f under certain conditions in the special case n 2 if the laurent expansion of f x around one singular point has only two consecutive homogeneous parts we show that s mathcal d f always admits an explicitly constructible semidefinite representation

obstructions to determinantal representability there has recently been ample interest in the question of which sets can be represented by linear matrix inequalities lmis a necessary condition is that the set is rigidly convex and it has been conjectured that rigid convexity is also sufficient to this end helton and vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices we disprove this conjecture by relating the question of finding lmi representations to the problem of determining whether a polymatroid is representable over the complex numbers we find a real zero polynomial such that no power of it admits a determinantal representation the proof uses recent results of wagner and wei on matroids with the half plane property and the polymatroids associated to hyperbolic polynomials introduced by gurvits

construction of unitary and normal companion matrices abstract for a polynomial f z a0 a1z an 1zn 1 zn a0 an 1 c with complex unimodular zeros we present two different methods to construct by a finite number of rational operations and square roots a unitary companion matrix i e a unitary matrix whose characteristic polynomial is 1 nf z we also show that rational operations alone cannot suffice finally we discuss the problem of constructing a normal companion matrix for a polynomial without any restrictions on its zeros and arrive at a negative answer

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

polynomial zerofinding iterative matrix algorithms abstract newbery s method is completed to a method for the construction of a complex symmetric or nonsymmetric matrix with a given characteristic polynomial the methods of fiedler schmeisser and dorfler and schmeisser for similar constructions of symmetric matrices are reviewed polynomials found in the literature are solved iteratively by one of fiedler s methods with initial values supplied either by schmeisser s method or taken on a large circle or randomly in a region of the complex plane the determinental equations are solved by the qr algorithm fiedler s method used iteratively exhibits fast convergence to simple roots even in the presence of multiple roots if at some iteration step the values of the iterates which are converging to a multiple root are averaged according to the hull mathon procedure then fast convergence is also attained for multiple roots this combination appears to have nice features for polynomials of small to moderate degree

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

mixed determinants and the kadison singer problem we adapt the arguments of marcus spielman and srivastava in their proof of the kadison singer problem to prove improved paving estimates working with anderson s paving formulation of kadison singer instead of weaver s vector balancing version we show that the machinery of interlacing polynomials due to marcus spielman and srivastava works in this setting as well the relevant expected characteristic polynomials turn out to be related to the so called mixed determinants that have been previously studied in a different context by borcea and branden this approach allows us to show that any projection with diagonal entries 1 2 can be 4 paved yielding improvements over the best known current estimates of 12 this approach also allows us to show that any projection with diagonal entries strictly less than 1 4 can be 2 paved matching recent results of bownik casazza marcus and speegle we also relate the problem of finding optimal paving estimates to bounding the root intervals of a natural one parameter deformation of the characteristic polynomial of a matrix that turns out to have several pleasing combinatorial properties

the algebraic degree of semidefinite programming given a generic semidefinite program specified by matrices with rational entries each coordinate of its optimal solution is an algebraic number we study the degree of the minimal polynomials of these algebraic numbers geometrically this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices we determine this degree using methods from complex algebraic geometry such as projective duality determinantal varieties and their chern classes

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

quantum convex support abstract convex support the mean values of a set of random variables is central in information theory and statistics equally central in quantum information theory are mean values of a set of observables in a finite dimensional c algebra a which we call quantum convex support the convex support can be viewed as a projection of the state space of a and it is a projection of a spectrahedron spectrahedra are increasingly investigated at least since the 1990s boom in semi definite programming we recall the geometry of the positive semi definite cone and of the state space we write a convex duality for general self dual convex cones this restricts to projections of state spaces and connects them to results on spectrahedra our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections using natural isomorphisms the result encodes the face lattice of the convex support into a set of projections in a and enables the integration of convex geometry with matrix calculus or algebraic techniques

singular points of the algebraic curves of symmetric hyperbolic forms abstract we investigate singular points of the algebraic curve defined by a hyperbolic form satisfying some symmetric properties we show that all singular points of such a curve are real nodes

containment problems for polytopes and spectrahedra we study the computational question of whether a given polytope or spectrahedron s a as given by the positive semidefiniteness region of a linear matrix pencil a x is contained in another s b first we classify the computational complexity extending results on the polytope polytope case by gritzmann and klee to the polytope spectrahedron case for various restricted containment problems np hardness is shown we then study in detail semidefinite conditions to certify containment building upon work by ben tal and nemirovski and helton klep and mccullough in particular we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron it is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases including containment of a spectrahedron in a polyhedron moreover in the case of bounded s a the criteria will always succeed in certifying containment of some scale

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

deciding polyhedrality of spectrahedra spectrahedra are linear sections of the cone of positive semidefinite matrices which as convex bodies generalize the class of polyhedra in this paper we investigate the problem of recognizing when a spectrahedron is polyhedral we generalize and strengthen results of m v ramana polyhedra spectrahedra and semidefinite programming in topics in semidefinite and interior point methods fields inst commun 18 ams providence ri 1998 pp 27 38 regarding the structure of spectrahedra and we devise a normal form of representations of spectrahedra this normal form is effectively computable and leads to an algorithm for deciding polyhedrality

remarks on the lax conjecture for hyperbolic polynomials abstract a conjecture of lax p lax differential equations difference equations and matrix theory commun pure appl math 11 1958 175 194 says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix the conjecture has recently been proved by lewis parillo ramana based on previous work of dubrovin and helton vinnikov in this note we prove related results for polynomials in several space variables which have rotational symmetries

construction of determinantal representation of trigonometric polynomials abstract for a pair of n n hermitian matrices h and k a real ternary homogeneous polynomial defined by f t x y det ti n xh yk is hyperbolic with respect to 1 0 0 the fiedler conjecture or lax conjecture is recently affirmed namely for any real ternary hyperbolic polynomial f t x y there exist real symmetric matrices s 1 and s 2 such that f t x y det ti n xs 1 ys 2 in this paper we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials

exposed faces of semidefinitely representable sets a linear matrix inequality lmi is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite motivated by the fact that diagonal lmis define polyhedra the solution set of an lmi is called a spectrahedron linear images of spectrahedra are called semidefinitely representable sets part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming such as one can do on polyhedra by linear programming it is known that every face of a spectrahedron is exposed this is also true in the general context of rigidly convex sets we study the same question for semidefinitely representable sets lasserre proposed a moment matrix method to construct semidefinite representations for certain sets our main result is that this method can work only if all faces of the considered set are exposed this necessary condition complements sufficient conditions recently proved by lasserre helton and nie

applications of stable polynomials to mixed determinants johnson s conjectures unimodality and symmetrized fischer products for n x n matrices a and b define eta a b sigma s det a s det b s where the summation is over all subsets of 1 n s is the complement of s and a s is the principal submatrix

positive polynomials and projections of spectrahedra this article is concerned with different aspects of spectrahedra and their projections sets that are important in semidefinite optimization we prove results on the limitations of so called lasserre and theta body relaxation methods for semialgebraic sets and varieties as a special case we obtain the main result of netzer plaumann and schweighofer siam j optim 20 2010 pp 1944 1955 on nonexposed faces we also solve the open problems from that work we further give a unified account of several results on convex hulls of curves and images of polynomial maps we finally prove a positivstellensatz for projections of spectrahedra which exceeds the known results that only work for basic closed semialgebraic sets

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

characteristic polynomials of symmetric matrices over the univariate polynomial ring abstract viewing a bivariate polynomial f r x t as a family of univariate polynomials in t parametrized by real numbers x we call f real rooted if this family consists of monic polynomials with only real roots if f is the characteristic polynomial of a symmetric matrix with entries in r x it is obviously real rooted in this article the converse is established namely that every real rooted bivariate polynomial is the characteristic polynomial of a symmetric matrix over the univariate real polynomial ring as a byproduct we present a purely algebraic proof of the helton vinnikov theorem which solved the 60 year old lax conjecture on the existence of definite determinantal representation of ternary hyperbolic forms

on the new variational principles and duality for periodic solutions of lagrange equations with superlinear nonlinearities periodic problem 1 1 was studied in the 1980s by many authors forthe sublinear as well as for the superlinear case see e g 6 however we believe that our paper may contribute some new look at this problem this is because we propose to study 1 1 by duality methods in a waythat is to some extent analogous to the methods developed for 1 1 insublinear cases 2 3 6 8 some cases of 1 1 for superlinear v

a new variational principle and duality for periodic solutions of hamilton s equations we are dealing with a classical boundary value problem concerning periodic solutions of hamiltonian equations p f k f 4th p f i f h x t p t 1 where h r x r x r r is a given function called the hamiltonian in mechanics 1 and also in mathematics 171 with a given hamiltonian the lagrangian l defined by the legendre transform l 4 x a sup k p wc x pi d 2

semidefinite descriptions of low dimensional separable matrix cones abstract let k e k e be convex cones residing in finite dimensional real vector spaces an element y in the tensor product e e is k k separable if it can be represented as finite sum y l x l x l where x l k and x l k for all l let s n h n q n be the spaces of n n real symmetric complex hermitian and quaternionic hermitian matrices respectively let further s n h n q n be the cones of positive semidefinite matrices in these spaces if a matrix a h mn h m h n is h m h n separable then it fulfills also the so called ppt condition i e it is positive semidefinite and has a positive semidefinite partial transpose the same implication holds for matrices in the spaces s m s n h m s n and for m 2 in the space q m s n we provide a complete enumeration of all pairs n m when the inverse implication is also true for each of the above spaces i e the ppt condition is sufficient for separability we also show that a matrix in q n s 2 is q n s 2 separable if and only if it is positive semidefinite

noncommutative polynomials nonnegative on a variety intersect a convex set abstract this paper gives a precise algebraic certificate for a noncommutative polynomial to be nonnegative on a free convex semialgebraic set intersect the variety of a free left ideal

on hyperboloidal cauchy data for vacuum einstein equations and obstructions to smoothness of scri the relationship between the geometric properties of hyperboloidal cauchy data for vacuum einstein equations at the conformal boundary of the initial data surface and between the space time geometry is analyzed in detail we prove that a necessary condition for existence of a smooth or a polyhomogeneous scri i e a scri around which the metric is expandable in terms ofr j log i r rather than in terms ofr j is the vanishing of the shear of the conformal boundary of the initial data surface we derive the boundary constraints which have to be satisfied by an initial data set for compatibility with friedrich s conformal framework we show that a sufficient condition for existence of a smooth scri not necessarily complete is the vanishing of the shear of the conformal boundary of the initial data surface and smoothness up to boundary of the conformally rescaled initial data we also show that the occurrence of some log terms in an asymptotic expansion at the conformal boundary of solutions of the constraint equations is related to the non vanishing of the weyl tensor at the conformal boundary

strict convexity of the joint c numerical range abstract let h h 1 h 2 h m be an m tuple of n n hermitian matrices and c c 1 c 2 c n r n the joint c numerical range w c h of h is defined for x x 1 x 2 x m r m the form associated with h is defined as f h t x det ti n x 1 h 1 x 2 h 2 x m h m we prove that the convex hull of w c h is strictly convex if the roots of f h t x 0 have constant multiplicities for all x on the unit sphere in r m the result is illustrated with several examples

polynomials compatible with a symmetric loewner matrix abstract several mutually equivalent conditions are proved for a polynomial of degree n and a symmetric n n matrix which form a basis for so called compatibility of these two objects one of the conditions yields a new parametric expression for a symmetric loewner matrix

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

cauchy problems for the conformal vacuum field equations in general relativity cauchy problems for einstein s conformal vacuum field equations are reduced to cauchy problems for first order quasilinear symmetric hyperbolic systems the hyperboloidal initial value problem where cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two surface is discussed and translated into the conformally related picture it is shown that for conformal hyperboloidal initial data of classhs s 4 there is a unique up to questions of extensibility development which is a solution of the conformal vacuum field equations of classhs it provides a solution of einstein s vacuum field equations which has a smooth structure at past null infinity

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

scalar curvature deformation and a gluing construction for the einstein constraint equations on a compact manifold the scalar curvature map at generic metrics is a local surjection f m we show that this result may be localized to compact subdomains in an arbitrary riemannian manifold the method is extended to establish the existence of asymptotically flat scalar flat metrics on n n 3 which are spherically symmetric hence schwarzschild at infinity i e outside a compact set such metrics provide cauchy data for the einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically schwarzschild near spatial infinity

a real symmetric tridiagonal matrix with a given characteristic polynomial abstract given a polynomial u x x n a n 1 x n 1 a 0 a v r v 0 1 n 1 having only real zeros we construct a real symmetric tridiagonal matrix whose characteristics polynomial is equal to 1 n u x this is a complete solution to a problem raised and partly solved by m fiedler

existence of periodic solutions for nonlinear differential equations abstract in this paper we shall study the existence of a periodic solution to the nonlinear differential equationxtt axt a xt a ax b t x f t x t 0 in some complex hilbert space using duality and variational methods

the multipole structure of stationary space times a definition of multipole moments for stationary asymptotically flat solutions of einstein s equations is proposed it is shown that these moments characterize a given space time uniquely conversely they can be arbitrarily prescribed i e they generate power series for the field variables which satisfy the field equations to all orders despite their apparently rather different origin they are shown to be identical with the geroch hansen ones

intertwining and testing matrices corresponding to a polynomial abstract given a natural number n and a polynomial of degree m n a description is given of a class r n of n by n matrices the intertwining matrices for with the following properties let h n be the family of all n by n hankel matrices compatible with and consider the relation hr t rh then an n by n matrix h belongs to h n if and only if this relation holds for all r r n a characterization is also given of testing matrices a matrix r is said to be a testing matrix if the fulfillment of the above relation already implies that h h n

roots of bivariate polynomial systems via determinantal representations we give two determinantal representations for a bivariate polynomial they may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two parameter eigenvalue problem the first determinantal representation is suitable for polynomials with scalar or matrix coefficients and consists of matrices with asymptotic order n 2 4 where n is the degree of the polynomial the second representation is useful for scalar polynomials and has asymptotic order n 2 6 the resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10 as well as for polynomials with a small number of terms

the wave equation on asymptotically de sitter like spaces in this paper we obtain the asymptotic behavior of solutions of the klein gordon equation on lorentzian manifolds x g which are de sitter like at infinity such manifolds are lorentzian analogues of the so called riemannian conformally compact or asymptotically hyperbolic spaces under global assumptions on the null bicharacteristic flow namely that the boundary of the compactification x is a union of two disjoint manifolds y and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to and to the other manifold as the parameter goes to we also define the scattering operator and show that it is a fourier integral operator associated to the bicharacteristic flow from y to y

symmetric determinantal representations in characteristic 2 abstract this paper studies symmetric determinantal representations sdr in characteristic 2 that is the representation of a multivariate polynomial p by a symmetric matrix m such that p det m and where each entry of m is either a constant or a variable we first give some sufficient conditions for a polynomial to have an sdr we then give a non trivial necessary condition which implies that some polynomials have no sdr answering a question of grenet et al a large part of the paper is then devoted to the case of multilinear polynomials we prove that the existence of an sdr for a multilinear polynomial is equivalent to the existence of a factorization of the polynomial in certain quotient rings we develop some algorithms to test the factorizability in these rings and use them to find sdrs when they exist altogether this gives us polynomial time algorithms to factorize the polynomials in the quotient rings and to build sdrs we conclude by describing the case of alternating determinantal representations in any characteristic

sturm and sylvester algorithms revisited via tridiagonal determinantal representations first we show that sturm algorithm and sylvester algorithm which compute the number of real roots of a given univariate polynomial lead to two dual tridiagonal determinantal representations of the polynomial next we show that the number of real roots of a polynomial given by a tridiagonal determinantal representation is greater than the signature of this representation

pencils of real symmetric matrices and real algebraic curves abstract let a rational real algebraic curve of degree n in the real projective plane x y z be given by its parametric equations let not containing the point o 0 0 1 have the property that i every line through the point o intersects in n real points counting multiplicities an explicit form is given for real symmetric matrices a and b of order n such that det xa yb zl 0 is an equation for for a diagonal b is a diagonal multiple of a loewner matrix the result partly proves a previous conjecture of the author about general real algebraic curves of degree n having the property i it is also shown that in this general case the conjecture is not true

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

smooth hyperbolicity cones are spectrahedral shadows hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials a well understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral in this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone that is a spectrahedral shadow

a randomized approximation algorithm for the minimal norm static output feedback problem a new randomized algorithm is suggested for extracting static output stabilizing feedbacks with approximately minimal norm for lti systems the algorithm has two similar stages where in the first one the feasibility problem is solved and in the second one the optimization problem is solved the formulation is unified for the feasibility and for the optimization problems as well as for continuous time or discrete time systems the method is demonstrated by applying it to the hard conjectured to be np hard problem of the minimal gain static output stabilizing feedback and to the hard conjectured to be np hard problem of regional pole placement via static output feedback in non convex or unconnected regions a proof of convergence in probability that captures the two rounds of the algorithm is given and complexity analysis is provided under some mild assumptions

determinantal representations of hyperbolic forms via weighted shift matrices we characterize hyperbolic ternary forms of degrees 3 4 and 5 that admit determinantal representations via cyclic weighted shift matrices

polynomial matrix inequality and semidefinite representation consider a convex set s x in cal d g x succeq 0 where g x is a symmetric matrix whose every entry is a polynomial or rational function d rn is a domain on which g x is defined and g x succeq 0 means g x is positive semidefinite the set s is called semidefinite representable if it equals the projection of a higher dimensional set that is defined by a linear matrix inequality lmi this paper studies sufficient conditions guaranteeing semidefinite representability of s we prove that s is semidefinite representable in the following cases i d rn g x is a matrix polynomial and matrix sos concave ii d is compact convex g x is a matrix polynomial and strictly matrix concave on d iii g x is a matrix rational function and q module matrix concave on d explicit constructions of semidefinite representations are given some examples are illustrated

maximum semidefinite and linear extension complexity of families of polytopes we relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise hausdorff distance of its members this result directly implies a known lower bound on the maximum semidefinite extension complexity of 0 1 polytopes we further show how our result can be used to improve on the corresponding bounds known for polygons with integer vertices our geometric proof builds upon nothing else than a simple well known property of maximum volume inscribed ellipsoids of convex bodies in particular it does not rely on factorizations over the semidefinite cone and thus avoids involved procedures of balancing them as required e g in briet et al math program 153 1 179 199 2015 moreover we show that the linear extension complexity of every d dimensional 0 1 polytope is bounded from above by o frac 2 d d

a composite polynomial zerofinding matrix algorithm abstract a globally convergent matrix algorithm is presented for finding the real and complex zeros of a complex polynomial p x a combination of schmeisser s and fiedler s matrices is used in the algorithm first the greatest common divisor g x of p x and its derivative p x is obtained and used to reduce p x to a polynomial q x p x q x with simple zeros second the zeros of q x are computed either by finding the eigenvalues of the first block of schmeisser s matrix or by applying fiedler s algorithm recursively in either case the eigenvalues are obtained by the qr algorithm third the multiplicity of each zero of p x is calculated by means of lagouanelle s modified limiting formula sample numerical results are given to demonstrate the effectiveness of the algorithm

periodic solutions of hamilton s equations and local minima of the dual action the dual action is a functional whose extremals lead to solutions of hamilton s equations up to now extremals of the dual action have been obtained either through its global minimization or through application of crit ical point theory a new methodology is introduced in which local minima of the dual action are found to exist applications are then made to the existence of hamiltonian trajectories having prescribed period 1 introduction a classical and very interesting boundary value problem is concerned with periodic solutions of hamilton s equations 1 1 p i vxtf x i p i x t vph x t p t where the function h rn x rn r is called the hamiltonian any such hamil tonian trajectory x p is such that h x t p t is constant this constant is called the energy associated with x p if x 0 x t p 0 p t for some t 0 then x p is said to have period t one can ask for what energies and periods are their associated hamiltonian trajectories this and related issues have both a long history and an active present we refer to desolneux and moulis 16 mancini 20 and rabinowitz 23 for surveys and a more extensive bibliography than would be appropriate here in this article attention will be focused upon the case in which the hamiltonian is a convex function a situation which has drawn perhaps the most attention of late we begin by reviewing briefly the basis for much of this recent work a dual action principle introduced by clarke in 1978 6 let g denote the conjugate of h in the sense of convex analysis i e the function

improved bounds in weaver and feichtinger conjectures we sharpen the constant in the ks2 conjecture of weaver 28 which was validated by marcus spielman and srivastava 26 in their solution of the kadison singer problem we then apply this result to prove optimal asymptotic bounds on the size of partitions in the feichtinger conjecture

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

an abstract theorem on the existence of periodic motions of non autonomous lagrange systems abstract in this paper we established an abstract theorem on the existence of periodic solutions for lagrange systems under tonelli framework strictly convex and superlinear completeness of phase flow which is based on the principle of least action in geometry and dynamics an example of the application of the abstract theorem is given and the existence of periodic solutions for a class of time periodic lagrange systems will be proved

duality for non convex variational principles abstract this paper treats the construction of dual variational principles for non convex problems using lagrangian methods besides the well known saddle point theory there are dualities involving minimizing the lagrangian in both variables and dualities where one first maximizes and then minimizes the lagrangian in each variable when the primal problem has certain structure an associated canonical lagrangian is defined which leads to the appropriate dual variational principle and to a hamiltonian description the properties of and correspondence between the critical points are analyzed the last five sections are devoted to examples of these dual principles

unitary similarity of the determinantal representation of unitary bordering matrices abstract let f x y z be a hyperbolic ternary form of degree n the helton vinnikov theorem asserts that there exists an n n complex symmetric matrix s such that f s x y z f x y z where the determinantal hyperbolic ternary form of an n n matrix b is defined by f b x y z det x b y b z i n let a be an n n unitary bordering matrix it is known that a is unitarily similar to a symmetric matrix in this paper we investigate the unitary similarity between the unitary bordering matrix a and the helton vinnikov symmetric matrix s admitted the determinantal representation of the ternary form f a x y z satisfying f s x y z f a x y z for n 3 4

periodic solutions to hamiltonian inclusions abstract we prove the existence of periodic trajectories of hamiltonian inclusions which reduce to the usual hamiltonian equations in the presence of smoothness this is accomplished via a direct variational principle involving a new action integral in an extended sense

expressing a polynomial as the characteristic polynomial of a symmetric matrix abstract we show that given a polynomial one can without knowing the roots construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed for a real polynomial the matrix can have purely imaginary off diagonal entries for a polynomial with only real distinct roots the matrix can be chosen real

two results on the size of spectrahedral descriptions a spectrahedron is a set defined by a linear matrix inequality given a spectrahedron we are interested in the question of the smallest possible size r of the matrices in the description by linear matrix inequalities we show that for the n dimensional unit ball r is at least frac n 2 if n 2 k 1 then we actually have r n the same holds true for any compact convex set in mathbb r n defined by a quadratic polynomial furthermore we show that for a convex region in mathbb r 3 whose algebraic boundary is smooth and defined by a cubic polynomial we have that r is at least five more precisely we show that if a b c in sym r mathbb r are real symmetric matrices such that f x y z det i r a x b y c z is a cubic polynomial then the surface in complex projective three space with affine equation f x y z 0 is singular

inverse power and durand kerner iterations for univariate polynomial root finding abstract univariate polynomial root finding is the oldest classical problem of mathematics and computational mathematics and is still an important research topic due to its impact on computational algebra and geometry the weierstrass durand kerner approach and its variations as well as matrix methods based on the qr algorithm are among the most popular practical choices for simultaneous approximation of all roots of a polynomial we propose an alternative application of the inverse power iteration to generalized companion matrices for polynomial root finding demonstrate its effectiveness and relate its study to unifying the derivation of the weierstrass durand kerner algorithm having quadratic convergence and its extensions having convergence rates 4 6 8 our experiments show substantial improvement versus the latter algorithm even though the inverse power iteration is most effective for the more limited tasks of approximating a single root or a few selected roots

nonlinear oscillations and boundary value problems for hamiltonian systems we prove the existence of solutions of various boundary value problems for nonautonomous hamiltonian systems with forcing terms begin gathered dot x t h p t x t p t g t hfill dot p t h x t x t p t f t hfill end gathered

reliability of perturbation theory in general relativity the relation between perturbation theory and exact solutions in general relativity is tackled by investigating the existence and properties of smooth one parameter families of solutions on the one hand the coefficients of the taylor expansion in the parameter of any given smooth family of solutions necessarily satisfy the hierarchy of perturbation equations on the other hand it is the converse question does any solution of the perturbation equations come from taylor expanding some family of exact solutions which is of importance for the mathematical justification of the use of perturbation theory this converse question is called the one of the reliability of perturbation theory using and completing recent results on the characteristic initial value problem the local reliability of perturbation theory for general relativity in vacuum is proven very generally this result is then generalized to the einstein yang mills equations and therefore in particular to the einstein maxwell ones