Determinantal Representations and Bézoutians

Abstract

We show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation. This is proved by considering sum-of-squares decompositions of certain bilinear forms called Bézoutians. Besides commutative algebra, the proof relies on results from real algebraic geometry.

v2: differently organized

References in corpus (3)

• Determinantal representations of hyperbolic plane curves: An elementary approach
• Hyperbolic polynomials, interlacers, and sums of squares
• A Note on the Hyperbolicity Cone of the Specialized Vámos Polynomial

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• Spectrahedral representations of plane hyperbolic curves
• Matroids on Eight Elements with the Half-plane Property and Related Concepts
• Positive Ulrich Sheaves