[Paper Review] Definite Determinantal Representations of Ternary Hyperbolic Forms

Christoph Hanselka|arXiv (Cornell University)|Nov 6, 2014

Advanced Differential Equations and Dynamical Systems7 references3 citations

TL;DR

This paper presents a fully algebraic, elementary proof of the Helton-Vinnikov theorem, establishing that every ternary hyperbolic polynomial admits a definite linear determinantal representation. By using symmetric $T$-spectral determinantal representations over affine curves and bounding matrix degrees via coefficient analysis, the authors show that $T$-real rooted polynomials in $\mathbb{R}[X,T]$ of bounded degree admit symmetric matrix representations with controlled entry degrees, thereby re-proving the Lax conjecture in three variables with algebraic methods.

ABSTRACT

We give a new and completely algebraic proof of the Helton-Vinnikov Theorem stating that every hyperbolic polynomial in three variables admits a definite linear determinantal representation.

Motivation & Objective

• To provide a purely algebraic, elementary proof of the Helton-Vinnikov theorem, avoiding advanced tools like theta functions on Jacobians.
• To establish that every ternary hyperbolic polynomial admits a linear determinantal representation that is definite at the hyperbolicity direction.
• To characterize the set of $T$-real rooted polynomials in $\mathbb{R}[X,T]$ of bounded degree and show they admit symmetric $T$-spectral determinantal representations with bounded matrix entry degrees.
• To replace the analytic and geometric methods of the original Helton-Vinnikov proof with algebraic techniques based on fractional ideals and degree bounds.

Proposed method

• Define $T$-real rootedness for bivariate polynomials and introduce symmetric $T$-spectral determinantal representations using monic characteristic polynomials of symmetric matrices over $\mathbb{R}[X]$.
• Use the map $\chi_{k,d}$ sending symmetric matrices of degree at most $k$ in $X$ to their characteristic polynomials in $\mathbb{R}[X,T]_{(k,d)}$, and prove its image is closed and contains all $T$-real rooted polynomials in this degree class.
• Establish boundedness of preimages under $\chi_{k,d}$ by analyzing coefficient bounds on $T$-roots and relating them to operator norms of symmetric matrices.
• Prove that boundedness of coefficients in $\mathbb{R}[T]_{\leq d}$ implies boundedness of matrix entries in $X$, using rescaling and compactness arguments on intervals $[0,1]$ and $(1,\infty)$.
• Use degree bounds from Lemma 25 to show that $T$-real rooted polynomials of total degree $d$ in $\mathbb{R}[X,Y,Z]$ admit linear determinantal representations with linear matrix entries.
• Apply the closedness of $\operatorname{im}\chi_{k,d}$ and density arguments to conclude that $H_{k,d} = \operatorname{im}\chi_{k,d}$, proving the existence of bounded-degree symmetric representations.

Experimental results

Research questions

• RQ1 Can the Helton-Vinnikov theorem be proven using only algebraic methods, without invoking theta functions or complex geometry?
• RQ2 For a $T$-real rooted polynomial in $\mathbb{R}[X,T]$ of bounded degree, does there exist a symmetric matrix representation with entries of controlled degree in $X$?
• RQ3 Is the image of the map $\chi_{k,d}$, sending symmetric matrices to their characteristic polynomials, closed in the space of $T$-real rooted polynomials of degree $d$ and bounded $X$-degree $k$?
• RQ4 Can the degree of matrix entries in a determinantal representation be bounded solely from coefficient bounds on the characteristic polynomial?
• RQ5 Does every ternary hyperbolic form admit a linear determinantal representation that is definite at the hyperbolicity direction, using only algebraic and analytic tools?

Key findings

• Every $T$-real rooted polynomial in $\mathbb{R}[X,T]$ of degree $d$ and bounded $X$-degree $k$ admits a symmetric $T$-spectral determinantal representation with matrix entries of degree at most $k$ in $X$.
• The image of the map $\chi_{k,d}$, which sends symmetric matrices to their characteristic polynomials, is closed in the space of $T$-real rooted polynomials of degree $d$ and $X$-degree at most $k$.
• The set $H_{k,d}$ of $T$-real rooted polynomials in $\mathbb{R}[X,T]_{(k,d)}$ is equal to $\operatorname{im}\chi_{k,d}$, proving the existence of bounded-degree symmetric representations.
• Boundedness of coefficients of $T$-polynomials implies boundedness of matrix entries in the corresponding symmetric representation, via norm and rescaling arguments.
• The Helton-Vinnikov theorem is reproven algebraically: every ternary hyperbolic polynomial admits a definite linear determinantal representation, with the proof relying only on degree bounds and algebraic topology of affine curves.
• The result holds over any real closed field, not just $\mathbb{R}$, due to the algebraic nature of the proof based on fractional ideals and class group divisibility.