Noncommutative polynomials nonnegative on a variety intersect a convex set

Abstract

Abstract. By a result of Helton and McCullough [HM12], 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 . Such polynomials must have the formwhere q i , r j are matrices of polynomials, and C j are real matrices satisfying C j L = LC j . 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 all ι ∈ I vanish if and only ifwhere each p i , q j and r k are matrices of polynomials of appropriate dimension, and each ι k is an element of the "L-real radical" of I. In this representation, we can restrict p i , q i , ι k and r k to be elements of a low-dimensional subspace of matrices of polynomials, and in particular, their degrees 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. This Positivstellensatz extends and unifies two different lines of results: (1) the free real Nullstellensatz of [CHMN13, Nel] which gives an algebraic certificate corresponding to one polynomial being zero on the free variety where others are zero; this is (P) with L = 1; (2) the convex Positivstellensatz of [HKM12, KS11] which is (P) without I; i.e., I = {0}. The representation (P) has a number of additional consequences which will be presented.

Finsler's Lemma for matrix polynomials

“…Similar results also exist for free polynomials, see [14], [15], [16]. (One-sided Real nullstellensatz for free polynomials is discussed in [9], [12], [19].)…”

Positivstellensätze for noncommutative rational expressions

“…Moreover, we note that the main result on positive rational functions, the noncommutative analogue of Artin's solution to Hilbert's seventeenth problem, that regular positive rational expressions are sums of squares [8], follows from our present theorem by taking an empty monic linear pencil, in fact, we obtain a slightly better matricial version of that result. Moreover, one has size bounds inherited from the Helton-Klep-Nelson convex Positivstellensatz [5], that is, checking that a noncommutative rational expression is effective using the algorithms given in [5].…”

The noncommutative Löwner theorem for matrix monotone functions over operator systems

“…We also note various noncommutative generalizations to the free functional calculus of Löwner's theorem were considered by the current author and Tully-Doyle [22], and by Palfia [19] previously, and to other functional calculi by Hansen [8] and Agler, M c Carthy, and Young [1]. Moreover, this work fits into a greater effort to systematize the theory of matrix inequalities [9,13,11,12,10].…”

“…It is clear that extension of f to the new domain must still be given by the same formula as before. Either using the algorithms in [10,21] or by brute force, one can see that…”

“…We also point out that the case where R 1 = R d as a diagonal algebra and R 2 = R was explored in [22,19], and that the current work simplifies the proof of the main result of those works if we are willing to use the commutative Löwner theorem from [1] as a black box. Moreover, if we are given a rational expression, such as the Schur complement, on a nice finite dimensional operator system, such as a matrix algebra, one can apply the algorithms in [10] which make the rational convex Positivstellensatz [21] effective to check that a function is matrix monotone in our sense.…”