(Even) Symmetric PSD and SOS forms

Abstract: In this talk we consider the so-called non-normalized limits of symmetric and even symmetric forms (homogeneous polynomials). To do so, we identify (even) symmetric forms of degree d for sufficiently many variables. The sets of positive semidefinite (non negative) and sums of squares of fixed degree form nested decreasing sequences under this identification. We completely characterize the question of non-negativity versus sums of squares in the non-normalized limit case. We begin by examining the symmetric quartics and provide test sets for non negativity and the property of being a sum of squares for the limit forms, and give interesting examples. Then, we consider even symmetric sextics and prove that the set of all psd limit forms is not semialgebraic and provide test sets as well (based on the work of Choi-Lam-Reznick). Finally, we study the tropicalizations of the duals to even symmetric psd and sos forms. Tropicalization reduces the study of even symmetric limit cones to the study of polyhedral cones. This is joint work together with Jose Acevedo, Greg Blekherman and Cordian Riener.

computational geometryalgebraic geometry

Audience: researchers in the topic

Aromath seminar

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