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Determinantal representations and Bézoutians

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Abstract

A major open question in convex algebraic geometry is whether all hyperbolicity cones are spectrahedral, i.e. the solution sets of linear matrix inequalities. We will use sum-of-squares decompositions of certain bilinear forms called Bézoutians to approach this problem. More precisely, we show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that \(q \cdot h\) has a definite determinantal representation. Besides commutative algebra, the proof relies on results from real algebraic geometry.

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Acknowledgments

This work is part of my Ph.D thesis. I would like to thank my advisor Claus Scheiderer for his encouragement and the Studienstiftung des deutschen Volkes for their financial and ideal support. I also thank Christoph Hanselka, Tim Netzer, Daniel Plaumann, Eli Shamovich, Bernd Sturmfels, Andreas Thom and Cynthia Vinzant for helpful discussions.

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  1. Fachbereich Mathematik und Statistik, Universität Konstanz, 78457, Konstanz, Germany

    Mario Kummer

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  1. Mario Kummer
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Correspondence to Mario Kummer.

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The author was supported by the Studienstiftung des deutschen Volkes.

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Kummer, M. Determinantal representations and Bézoutians. Math. Z. 285, 445–459 (2017). https://doi.org/10.1007/s00209-016-1715-9

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