Noncommutative polynomials nonnegative on a variety intersect a convex set

TL;DR

This paper establishes a precise algebraic certificate, called a 'Perfect' Positivstellensatz, for polynomials nonnegative on convex semialgebraic sets intersecting a variety, with applications to defining polynomials and radical computations.

Contribution

It introduces a new algebraic certificate for nonnegativity on convex semialgebraic sets intersecting a variety and provides an efficient algorithm for computing the L-real radical.

Findings

Characterization of polynomials positive on convex free semialgebraic sets

Explicit sum of squares representation involving the L-real radical

Efficient algorithm for radical computation

Abstract

By a result of Helton and McCullough, open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D_L of a linear matrix inequality (LMI) L(X)>0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called "Perfect" Positivstellensatz. For example, given a generic convex free semialgebraic set D_L we determine all "(strong sense) defining polynomials" p for D_L. This follows from our general result for a given linear pencil L and a finite set I of rows of polynomials. A matrix polynomial p is positive where L is positive and I vanishes if and only if p has a weighted sum of squares representation module the "L-real radical" of I. In such a representation the degrees of the polynomials appearing depend in a very tame way only on the degree of p and the degrees of the elements of I. Further, this paper gives an efficient algorithm for computing the L-real radical of I. Our Positivstellensatz has a number of additional consequences which are presented.