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Hyperbolic polynomials, interlacers, and sums of squares

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Abstract

Hyperbolic polynomials are real polynomials whose real hypersurfaces are maximally nested ovaloids, the innermost of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. Here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it. This set of interlacers is a convex cone, which we write as a linear slice of the cone of nonnegative polynomials. In particular, this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials. Using a sums of squares relaxation, we then approximate a hyperbolicity cone by the projection of a spectrahedron. A multiaffine example coming from the Vámos matroid shows that this relaxation is not always exact. Using this theory, we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists.

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Acknowledgments

We would like to thank Alexander Barvinok, Petter Brändén, Tim Netzer, Rainer Sinn, and Victor Vinnikov for helpful discussions on the subject of this paper. Daniel Plaumann was partially supported by a Feodor Lynen return fellowship of the Alexander von Humboldt-Foundation. Cynthia Vinzant was partially supported by the National Science Foundation RTG grant DMS-0943832 and award DMS-1204447.

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Authors and Affiliations

  1. Universität Konstanz, Constance, Germany

    Mario Kummer & Daniel Plaumann

  2. University of Michigan, Ann Arbor, MI, USA

    Cynthia Vinzant

Authors
  1. Mario Kummer
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  2. Daniel Plaumann
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  3. Cynthia Vinzant
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Correspondence to Daniel Plaumann.

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Kummer, M., Plaumann, D. & Vinzant, C. Hyperbolic polynomials, interlacers, and sums of squares. Math. Program. 153, 223–245 (2015). https://doi.org/10.1007/s10107-013-0736-y

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  • DOI: https://doi.org/10.1007/s10107-013-0736-y

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