Kolloquien und Oberseminare Übersicht Detail

Vortragende Person/Vortragende Personen:
Aljaž Zalar

In 2006, Blekherman established estimates on the sizes of the cones of nonnegative forms and sums of squares forms by comparing the volume radii of compact sections of these cones with a suitably chosen hyperplane. For a fixed degree bigger than 2, as the number of variables goes to infinity, the ratio between the volume radii goes to 0.
In the talk we will present estimates for the ratio between the volume radii of nonnegative forms and sums of squares forms in three different vector spaces: 1) the space of biquadratic biforms, 2) the space of biquadratic biforms modulo the ideal of all orthonormal 2–frames, 3) the space of even quartic forms. In 1) and 2), the conclusion is analogous as in the case of all forms, while in 3), slightly surprisingly, the difference between the cones in question does not grow arbitrarily large as the number of variables grows to infinity. The motivation to study the gap between positive polynomials and sums of squares polynomials in 1) comes from quantum information theory, in 2) from financial mathematics, and in 3) from matrix theory. In 1) the gap corresponds to the gap between positive maps and completely positive maps between matrix algebras, in 2) to the gap between cross–positive maps and completely cross–positive maps between matrix algebras, and in 3) to the gap between copositive matrices and matrices that are a sum of a positive semidefinite matrix and entrywise nonnegative one.

This is joint work with Igor Klep, Scott McCullough, Klemen Sivic and Tea Strekelj.