Non-commutative polynomials with convex level slices

TL;DR

This paper characterizes symmetric non-commutative polynomials with convex level sets in matrix variables, showing they are at most quadratic in certain variables under mild conditions, with applications in control theory and matrix convexity.

Contribution

It provides an algebraic certificate for such polynomials, establishing their degree bound and convexity properties in a non-commutative setting.

Findings

Polynomials with convex level sets are at most quadratic in certain variables.

The paper offers an algebraic certificate characterizing these polynomials.

Applications include control system design and matrix convexity theories.

Abstract

Let a and x denote tuples of (jointly) freely noncommuting variables. A square matrix valued polynomial p in these variables is naturally evaluated at a tuple (A,X) of symmetric matrices with the result p(A,X) a square matrix. The polynomial p is symmetric if it takes symmetric values. Under natural irreducibility assumptions and other mild hypothesis, the article gives an algebraic certificate for symmetric polynomials p with the property that for sufficiently many tuples A, the set of those tuples X such that p(A,X) is positive definite is convex. In particular, p has degree at most two in x. The case of noncommutative quasi-convex polynomials is of particular interest. The problem analysed here occurs in linear system engineering problems. There the A tuple corresponds to the parameters describing a system one wishes to control while the X tuple corresponds to the parameters one seeks in designing the controller. In this setting convexity is typically desired for numerical reasons and to guarantee that local optima are in fact global. Further motivation comes from the theories of matrix convexity and operator systems.