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Let S = {x ∈ R n | g 1 (x) ≥ 0,. .. , gm(x) ≥ 0} be a basic closed semialgebraic set defined by real polynomials g i. Putinar's Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = P m i=0 σ i g i where g 0 := 1 and each σ i is a sum of squares of… (More)

We show that all the coefficients of the polynomial tr((A + tB) m) ∈ R[t] are nonnegative whenever m ≤ 13 is a nonnegative integer and A and B are positive semidefinite matrices of the same size. This has previously been known only for m ≤ 7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani… (More)

A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g 1 ≥ 0,. .. , gm ≥ 0. We give a short introduction to Lasserre's method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as… (More)

Schmüdgen's Positivstellensatz roughly states that a polynomial f positive on a compact basic closed semialgebraic subset S of R n can be written as a sum of polynomials which are nonnegative on S for certain obvious reasons. However, in general, you have to allow the degree of the summands to exceed largely the degree of f. Phenomena of this type are one… (More)

We present a new proof of Schmüdgen's Positivstellensatz concerning the representation of polynomials f ∈ R[X 1 , ..., X d ] that are strictly positive on a compact basic closed semialgebraic subset S of R d. Like the two other existing proofs due to Schmüdgen and Wörmann, our proof also applies the classical Positivstellensatz to non–constructively produce… (More)

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly in-feasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the… (More)

We consider the problem of computing the global infimum of a real polynomial f on R n. Every global minimizer of f lies on its gradient variety, i.e., the algebraic subset of R n where the gradient of f vanishes. If f attains a minimum on R n , it is therefore equivalent to look for the greatest lower bound of f on its gradient variety. Nie, Demmel and… (More)

We show that Connes' embedding conjecture on von Neumann algebras is equivalent to the existence of certain algebraic certificates for a polynomial in noncommuting variables to satisfy the following nonnegativity condition: The trace is nonnegative whenever self-adjoint contraction matrices of the same size are substituted for the variables. These algebraic… (More)

We prove a criterion for an element of a commutative ring A to be contained in an archimedean semiring T ⊂ A. It can be used to investigate the question whether nonnegativity of a polynomial on a compact semialgebraic set can be certified in a certain way. In case of (strict) positivity instead of nonnegativity, our criterion simplifies to classical results… (More)

In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict posi-tivity of a polynomial f on a basic closed set K ⊂ R n. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean… (More)