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On hyperbolicity cones associated with elementary symmetric polynomials

Yuriy Zinchenko(zinchen***at***mcmaster.ca)

Abstract: Elementary symmetric polynomials can be thought of as derivative polynomials of $E_n(x)=\prod_{i=1,\ldots,n} x_i$. Their associated hyperbolicity cones give a natural sequence of relaxations for $\Re^n_+$. We establish a recursive structure for these cones, namely, that the coordinate projections of these cones are themselves hyperbolicity cones associated with elementary symmetric polynomials. As a consequence of this recursion, we give an alternative characterization of these cones, and give an algebraic characterization for one particular dual cone associated with $E_{n-1}(x)=\sum_{1 \leq i \leq n} \prod_{j \neq i} x_j$ together with its self-concordant barrier functional.

Keywords: hyperbolic polynomials; hyperbolicity cones; elementary symmetric polynomials; positive semi-definite representability

Category 1: Linear, Cone and Semidefinite Programming (Other )

Citation:

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Entry Submitted: 09/04/2007
Entry Accepted: 09/04/2007
Entry Last Modified: 09/04/2007

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