Positive polynomials in scalar and matrix variables, the spectral theorem and optimization

TL;DR

This paper surveys the historical development and recent advances in positive polynomials, matrix variables, and their applications in optimization, control theory, and algebraic geometry, highlighting new connections and research directions.

Contribution

It provides a comprehensive overview of the interplay between positivity, sums of squares, and optimization in matrix and polynomial contexts, including recent developments in free *-algebras.

Findings

Connections between real algebraic geometry and optimization are expanding.

Recent studies reveal new structures of positivity and convexity in free *-algebras.

Applications to control theory demonstrate practical relevance of polynomial positivity.

Abstract

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free *-algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.