Positive polynomials in matrix variables

Abstract: Hilbert's 17th problem asked whether every positive polynomial can be written as a quotient of sums of squares of polynomials. As many others on Hilbert's famous list, this problem and its affirmative resolution by Emil Artin started a thriving mathematical discipline, known as real algebraic geometry. At its core, it studies the interplay between polynomial inequalities and positivity (geometry) and sums of squares certifying such positivity (algebra). Apart from its pure mathematics appeal, this theory is the pillar of polynomial optimization, since sums of squares can be efficiently traced via semidefinite programming.

This talk reviews old and new results on positivity of noncommutative polynomials and their traces, in terms of their matrix evaluations. There are two natural setups to consider: positivity in matrix variables of a given fixed size, and positivity in matrix variables of arbitrary size. This talk compares the sums-of-squares certificates of positivity across these two setups, their shortcomings and open ends.

commutative algebraalgebraic geometrycombinatoricsrings and algebrasrepresentation theory

Audience: researchers in the topic

Geometry, Algebra, Singularities, and Combinatorics

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