Real Algebraic Geometry For Matrix Variables

In this project we will derive new classes of noncommutative Positivstellensätze (= positivity certificates) and investigate the gap between positive polynomials and sums of squares.  Ever since Gauss it was known that a positive univariate real polynomial can be written as a sum of two squares of real polynomials. In a similar spirit, positive semidefinite quadratic forms (in any number of variables) are sums of squares of linear forms. During his 1885 PhD thesis defense Minkowski got into an argument with Hilbert about whether an extension to higher degree forms holds true (i.e., is every positive polynomial a sum of squares (sos) of polynomials?), thus providing a common generalization of the above two observations. A few years later Hilbert answered this in the negative; his proof was highly non-constructive, and the first explicit example of a positive polynomial that is not sos was given only 80 years later by Motzkin in 1967. Hilbert also posited that positive bivariate polynomials are sums of squares of rational functions, leading him to include the following problem (as #17) among the famous 23 problems for his address to the 1900 International Congress of Mathematicians: (H17) Is every positive polynomial a sum of squares of rational functions? A positive solution was presented in 1926 by Artin who developed the theory of formally real fields to solve this problem; we call this the beginning of real algebraic geometry (RAG). Nowadays real algebraic geometry (see, e.g., Bochnak, Coste, Roy 1998) is the branch of algebraic geometry studying real algebraic sets, i.e., real-number solutions to systems of polynomial equations. Pillars of RAG are generalizations of the above mentioned theorem of Artin, the so-called Positivstellensätze (=certificates of positivity): (¿P.Satz?) Given polynomials p and q, is p positive where q is positive? Also of interest are their off-shoot, Nullstellensätze (does p vanish whenever q vanishes?). RAG has seen rapid growth in recent years fueled by its wide range of applications in areas such as statistics, economics, and computer science (see, e.g., Henrion, Korda, Lasserre 2021). On the other hand, many problems in linear systems design in control theory (Skelton, Iwasaki, Grigoriadis 1997) or quantum physics (Navascués, Pironio, Acín 2010) have matrices as variables, and the formulas naturally contain noncommutative polynomials in matrices. Analyzing such problems has led to the development recently of a noncommutative (nc) real algebraic geometry. Often, the qualitative properties of the noncommutative case are much cleaner than those of their scalar counterparts. Indeed, the relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure. As in the classical case, the Nullstellensatz and Positivstellensatz are pillars of nc RAG. This proposal aims to develop new nc Positivstellensätze and tools and techniques to (Δ) determine and analyze the gap between positive polynomials and sums of squares or the various natural sos extensions arising from Positivstellensätze. Our focus will be on problems in matrix unknowns. Recent advances in real algebraic geometry and noncommutative algebra yield exciting new approaches to these problems, but there are fundamental challenges ahead. The proposal intends to overcome these by providing and using novel algebraic, geometric and analytic tools.