---
abstract: |
    Hilbert proved that a non-negative real quartic form $f(x,y,z)$ is the sum of three squares of quadratic forms.  We give a new proof which shows that if the plane curve $Q$ defined by $f$ is smooth, then $f$ has exactly $8$ such representations, up to equivalence.  They correspond to those real $2$-torsion points of the Jacobian of $Q$ which are not represented by a conjugation-invariant divisor on $Q$.

    0.5
address:
- 'Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA'
- 'Department of Mathematics, University of Illinois, Urbana, IL 61801, USA'
- ' Fachbereich für Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Allemagne '
- 'Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA'
author:
- Victoria Powers
- Bruce Reznick
- Claus Scheiderer
- Frank Sottile
title: 'A New Proof of Hilbert’s Theorem on Ternary Quartics'
---

Introduction
============

A *ternary quartic* is a homogeneous polynomial $f(x,y,z)$ of degree $4$ in three variables.  If $f$ has real coefficients, then $f$ is *non-negative* if $f(x,y,z)\ge0$ for all real $x$, $y$, $z$. Hilbert [@Hi1888] showed that every non-negative real ternary quartic form is a sum of three squares of quadratic forms.  His proof (see [@Rudin], [@Swan] for modern expositions) was non-constructive: The map $$\pi\colon\ (p,q,r)\ \longmapsto\ p^2+q^2+r^2$$ from triples of real quadratic forms to non-negative quartic forms is surjective, as it is both open and closed when restricted to the preimage of the (dense) connected set of non-negative quartic forms which define a smooth complex plane curve.  An elementary and constructive approach to Hilbert’s theorem was recently begun by Pfister [@Pf04]. 

A *quadratic representation* of a complex ternary quartic form $f=f(x,y,z)$ is an expression $$\label{E:Q-rep}
   f\ =\ p^2 + q^2 + r^2$$ where $p$, $q$, $r$ are complex quadratic forms.  A representation $f=(p')^2 + (q')^2 + (r')^2$ is *equivalent* to this if $p,q,r$ and $p',q',r'$ have the same linear span in the space of quadratic forms. 

Powers and Reznick [@PR00] investigated quadratic representations computationally, using the Gram matrix method of [@CLR].  In several examples of non-negative ternary quartics, they always found $63$ inequivalent representations as a sum of three squares of complex quadratic forms; $15$ of these were sums or differences of squares of real forms.  We explain these numbers, in particular the number $15$, and show that precisely $8$ of the $15$ are sums of squares. 

If the complex plane curve $Q$ defined by $f=0$ is smooth, it has genus $3$, and so the Jacobian $J$ of $Q$ has $2^6-1=63$ non-zero $2$-torsion points.  Coble [@Co1929] showed that these are in one-to-one correspondence with equivalence classes of quadratic representations of $f$. If $f$ is real, then $Q$ and $J$ are defined over ${{\mathbb{R}}}$. The non-zero $2$-torsion points of $J({{\mathbb{R}}})$ correspond to *signed quadratic representations* $f=\pm p_1\pm p_2\pm p_3$, where $p_i$ are real quadratic forms.  If $f$ is also non-negative, the real Lie group $J({{\mathbb{R}}})$ has two connected components, and hence has $2^4-1=15$ non-zero $2$-torsion points.  We use Galois cohomology to determine which $2$-torsion points give rise to sum of squares representations over ${{\mathbb{R}}}$.

\[T:main\] Suppose that $f(x,y,z)$ is a non-negative real quartic form which defines a smooth complex plane curve $Q$. Then the inequivalent representations of $f$ as a sum of three squares are in one-to-one correspondence with the eight $2$-torsion points in the non-identity component of $J({{\mathbb{R}}})$, where $J$ is the Jacobian of $Q$.

Wall [@Wa91] studies quadratic representations of (possibly singular) complex ternary quartic forms $f$. Again, in the irreducible case, the non-trivial $2$-torsion points on the generalized Jacobian give equivalence classes of quadratic representations of $f$. These representations are special in that they have no basepoints. 

Quadratic representations with a given base locus $B$ correspond to the $2$-torsion points on the Jacobian of a curve ${\widetilde{Q}}$, which is the image of $Q$ under the complete linear series of quadrics through $B$. Classifying all possibilities for $B$ gives the number of inequivalent quadratic representations of $f$. If $f$ is real and non-negative, this classification, together with arguments from Galois cohomology, gives all inequivalent representations of $f$ as a sum of squares.  This complete analysis will appear in an unabridged version. 

We thank C.T.C. Wall, who brought his work to our attention, and the organizers of the RAAG conference in Rennes in June 2001, where this work began. 

Basepoint-free quadratic representations {#S:PFQR}
========================================

Let $f(x,y,z)$ be an irreducible quartic form over ${{\mathbb{C}}}$, and let $Q$ be the complex plane curve $f=0$. The Picard group $\operatorname{Pic}(Q)$ of $Q$ is the group of Weil divisors on the regular part of $Q$, modulo divisors of rational functions which are invertible around the singular locus of $Q$. Let $J_Q$ be the generalized Jacobian of $Q$, so that $J_Q({{\mathbb{C}}})$ is the identity component of $\operatorname{Pic}(Q)$. Its structure may be determined from the Jacobian of the normalization ${\widetilde{Q}}$ of $Q$ via the exact sequence [@Ha77 Ex.  II.6.9] $$0\ \longrightarrow\
    \bigoplus_{p\in Q} {\widetilde{{{\mathcal{O}}}}}^*_p/{{\mathcal{O}}}^*_p
    \ \longrightarrow\ J_Q({{\mathbb{C}}})\ \longrightarrow\
    J_{{\widetilde{Q}}}({{\mathbb{C}}})\ \longrightarrow\ 0\,,$$ where ${{\mathcal{O}}}_p$ is the local ring of $Q$ at $p$, ${\widetilde{{{\mathcal{O}}}}}_p$ is its normalization, and $^*$ indicates the group of units. 

The *base locus* $B$ of a quadratic representation  of $f$ is the zero scheme of the homogeneous ideal generated by the forms $p$, $q$, $r$. The closed subscheme $B$ is supported on the singular locus of $Q$. We say that is *basepoint-free* if $B$ is empty. 

The non-trivial $2$-torsion points of $J_Q$ are in one-to-one correspondence with the equivalence classes of basepoint-free quadratic representations of $f$.

[*Proof.*]{}  Given a basepoint-free quadratic representation , consider the map $$\varphi\colon{{\mathbb{P}}}^2\to{{\mathbb{P}}}^2,\quad x\mapsto\bigl(p(x):q(x):r(x)
\bigr).$$ The image of $Q$ under $\varphi$ is the conic $C$ defined by the equation $y_0^2+y_1^2+y_2^2=0$. Let $y$ be a point in $C$ whose preimages are regular points of $Q$. Then $\varphi^*(y)$ is an effective divisor of degree $4$ that is not the divisor of a linear form.  Indeed, after a linear change of coordinates we can assume $y=
(0:1:i)$. A linear form vanishing on $\varphi^*(y)$ would divide each conic $\alpha p+\beta(q+ir)$ through $\varphi^*(y)$, and thus would divide $$f\ =\ p^2+(q+ir)(q-ir)\,,$$ contradicting the irreducibility of $f$.

Fix a linear form $\ell$ that does not vanish at any singular point of $Q$. Then $L:={{\rm div}}(\ell)$ is an effective divisor of degree $4$ on $Q$. Let $\zeta=[\varphi^*(y)-L]$. Since $2y$ is the divisor of a linear form (the tangent line to $C$ at $y$), $\varphi^*(2y)$ is the divisor on $Q$ of a quadratic form.  Thus $2\zeta=0$. Moreover, $\zeta
\ne0$ as $\varphi^*(y)$ is not the divisor of a linear form.  The $2$-torsion point $\zeta$ of $J_Q$ depends only upon the map $\varphi$.

Conversely, suppose that $\zeta\in J_Q({{\mathbb{C}}})$ is a non-zero $2$-torsion point.  Let $D\ne D'$ be effective divisors which represent the class $\zeta+[L]$ in $\operatorname{Pic}(Q)$. As $Q$ has arithmetic genus 3, the Riemann-Roch Theorem implies that there is a pencil of such divisors.  Then $2D$, $2D'$ and $D+D'$ are effective divisors of degree $8$, and are all linearly equivalent to $2L$, the divisor of a conic.  By the Riemann-Roch Theorem there are quadratic forms $q_0$, $q_1$ and $q_2$ such that

${{\rm div}}(q_0)=2D$, ${{\rm div}}(q_1)=2D'$ and ${{\rm div}}(q_2)=D+D'$.

Therefore, the rational function $g:=q_0q_1/q_2^2$ on $Q$ is constant.  Scaling $q_1$ and $q_2$ appropriately, we may assume that $g\equiv1$ on $Q$ and also that $f=q_0q_1-q_2^2$. Diagonalizing the quadratic form $q_0q_1-q_2^2$ gives a quadratic representation for $f$. This defines the inverse of the previous map. 

Quadratic representations of real quartics
==========================================

Suppose now that $f$ is a non-negative real quartic form defining a real plane curve $Q$ with complexification $Q_{{\mathbb{C}}}=Q\otimes_{{\mathbb{R}}}{{\mathbb{C}}}$. The elements of $\operatorname{Pic}(Q)$ can be identified with those divisor classes in $\operatorname{Pic}(Q_{{\mathbb{C}}})$ that are represented by a conjugation-invariant divisor.  Let $J$ be the generalized Jacobian of $Q$.

If $\zeta\in J({{\mathbb{C}}})$ is the $2$-torsion point corresponding to a signed quadratic representation $$f\ =\ \pm p^2\pm q^2 \pm r^2$$ consisting of real polynomials $p$, $q$, $r$, then $\zeta={\overline{\zeta}}$, i.e., $\zeta\in J({{\mathbb{R}}})$.

Conversely, let $0\ne\zeta\in J({{\mathbb{R}}})$ with $2\zeta=0$. Choose a real linear form $\ell$ not vanishing on the singular points of $Q$, and let $L={{\rm div}}(\ell)$. We can choose effective divisors $D\ne{\overline{D}}$ on $Q_{{\mathbb{C}}}$ representing the class $\zeta+[L]$. Then $2D$, $2{\overline{D}}$ and $D+{\overline{D}}$ are each equivalent to $2L$. Let $r$ be a real quadratic form with divisor $D+{\overline{D}}$, and let $g$ be a (complex) quadratic form with divisor $2D$ (both divisors taken on $Q_{{\mathbb{C}}}$).  Since $D\sim{\overline{D}}$, there is a rational function $h$ on $Q_{{\mathbb{C}}}$, invertible around $Q_{{\rm sing}}$, with ${{\rm div}}(h)=D-{\overline{D}}$. Let $c=h{\overline{h}}$, a nonzero real constant on $Q$. Since ${{\rm div}}(r)={{\rm div}}(g)+{{\rm div}}(h)$, there is a complex number $\alpha\ne0$ with $\frac rg=\alpha h$ on $Q$, which implies that $$c|\alpha|^2\ =\ \frac rg\>\frac{{\overline{r}}}{{\overline{g}}}\ =\ \frac{r^2}{p^2+q^2}$$ on $Q$, where $p,q$ are the real and imaginary parts of $g=p+iq$. So the quartic form $$u\ :=\ r^2-c\,|\alpha|^2(p^2+q^2)$$ vanishes identically on $Q$. Since $u\ne0$, $f$ is a constant multiple of $u$. If $c>0$, we get a signed quadratic representation of $f$, with both signs $\pm$ occuring.  If $c<0$, $f$ must be a positive multiple of $u$ since $f$ is non-negative, and we get a representation of $f$ as a sum of three squares of real forms. 

We now calculate the sign of $c$. For this we use the exact sequence $$\label{HS}
   0\to\operatorname{Pic}(Q)\to\operatorname{Pic}(Q_{{\mathbb{C}}})^G{\overset{\partial}{\longrightarrow}}\operatorname{Br}({{\mathbb{R}}})\to H^2_{{\textnf{\'et}}}(Q,{{\mathbb{G}}}_m)$$ of étale cohomology groups.  It arises from the Hochschild-Serre spectral sequence for the Galois covering $Q_{{\mathbb{C}}}\to Q$ and coefficients ${{\mathbb{G}}}_m$. Here $G=\operatorname{Gal}({{\mathbb{C}}}/{{\mathbb{R}}})$ acts on $\operatorname{Pic}(Q_{{\mathbb{C}}})$ by conjugation, and $\operatorname{Pic}(Q_{{\mathbb{C}}})^G$ is the group of $G$-invariant divisor classes.  Moreover, $\operatorname{Br}({{\mathbb{R}}})=H^2_{{\textnf{\'et}}}(\operatorname{Spec}{{\mathbb{R}}},{{\mathbb{G}}}_m)$ is the Brauer group of ${{\mathbb{R}}}$ (which is of order $2$), and $\operatorname{Br}({{\mathbb{R}}})\to H^2_{{\textnf{\'et}}}(Q,{{\mathbb{G}}}_m)$ is the restriction map. 

It is easy to see that $c<0$ if and only if $\partial(\zeta)$ is the non-trivial class in $\operatorname{Br}({{\mathbb{R}}})$. If $Q$ has an ${{\mathbb{R}}}$-point, then $\operatorname{Br}({{\mathbb{R}}})\to H^2_{{\textnf{\'et}}}(Q,{{\mathbb{G}}}_m)$ has a splitting given by that point, and hence $\partial$ vanishes identically. 

If $Q$ is smooth, then $f$ non-negative forces $Q({{\mathbb{R}}})=\emptyset$, and the map $\operatorname{Br}({{\mathbb{R}}})\to H^2_{{\textnf{\'et}}}(Q,{{\mathbb{G}}}_m)$ is zero.  In this case, $Pic(Q_C)^G$ contains an odd degree divisor if and only if the genus of $Q$ is even and $J({{\mathbb{R}}})^0$, the identity connected component of the real Lie group $J({{\mathbb{R}}})$, is the kernel of the restriction $J({{\mathbb{R}}})\to\operatorname{Br}({{\mathbb{R}}})$ of $\partial$ [@We1883; @Ge66].  Since in our case $g(Q)=3$, this implies that the sequence $$0\to J({{\mathbb{R}}})^0\to J({{\mathbb{R}}}){\overset{\partial}{\longrightarrow}}\operatorname{Br}({{\mathbb{R}}})\to0$$ is (split) exact.  If $Q$ is singular with $Q({{\mathbb{R}}})=\emptyset$, one compares sequence for $Q$ to the same sequence for the normalization ${\widetilde{Q}}$ of $Q$ and concludes that $\partial\colon J({{\mathbb{R}}})\to\operatorname{Br}({{\mathbb{R}}})$ is surjective as well.  We complete the proof of Theorem \[T:main\].  Since $f$ is non-negative and $Q$ is smooth of genus $3$, $J({{\mathbb{R}}})^0\cong (S^1)^3$ as a real Lie group.  By the facts just mentioned, there exist $2^4-1=15$ non-zero $2$-torsion elements in $J({{\mathbb{R}}})$. The $8$ that do not lie in $J({{\mathbb{R}}})^0$, or equivalently, which cannot be represented by a conjugation-invariant divisor on $Q_{{\mathbb{C}}}$, are precisely those that give rise to sums of squares representations of $f$.

[10]{}

M. D. Choi, T. Y. Lam, B. Reznick: Sums of squares of real polynomials.  In: *$K$-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras* (Santa Barbara, 1992), Proc.  Symp.  Pure Math.  **58**, Am.  Math.  Soc.,  Providence, RI, 1995, pp.  103–126.  A. B. Coble: *Algebraic Geometry and Theta Functions*.  Am.  Math.  Soc.  Colloquium Publications **10**, Am.  Math. Soc.,  1929. 

W.-D. Geyer: Ein algebraischer Beweis des Satzes von Weichold über reelle algebraische Funktionenkörper.  In: Algebraische Zahlentheorie (Oberwolfach, 1964), H. Hasse and P. Roquette (eds.),  Mannheim, 1966, pp.  83–98. 

R. Hartshorne: *Algebraic Geometry*.  Grad.  Texts Math.  **52**, Springer, New York, 1977.  D. Hilbert: Über die Darstellung definiter Formen als Summe von Formenquadraten.  Math.  Ann.  **32**, 342–350 (1888). 

A. Pfister: On Hilbert’s theorem about ternary quartics.  In: *Algebraic and Arithmetic Theory of Quadratic Forms*, Contemp.  Math. ** 344**, Am.  Math.  Soc.,  Providence, RI, 2004. 

V. Powers, B. Reznick: Notes towards a constructive proof of Hilbert’s theorem on ternary quartics.  In: *Quadratic Forms and Their Applications* (Dublin, 1999), Contemp.  Math.  **272**, Am.  Math.  Soc.,  Providence, RI, 2000, pp.  209–227.  W. Rudin: Sums of squares of polynomials.  Am.  Math.  Monthly **107**, 813–821 (2000). 

R. G. Swan: Hilbert’s theorem on positive ternary quartics.  In: *Quadratic Forms and Their Applications* (Dublin, 1999), Contemp.  Math.  **272**, Am.  Math.. Soc.,  Providence, RI, 2000, pp.  287–292.  C. T. C. Wall: Is every quartic a conic of conics?  Math.  Proc.  Cambridge Phil.  Soc.  **109**, 419–424 (1991).  G. Weichold: Über symmetrische Riemannsche Flächen und die Periodizitätsmoduln der zugehörigen Abelschen Normalintegrale erster Gattung.  Zeitschr.  Math.  Phys.  **28**, 321–351 (1883).