295 research outputs found
Positive trace polynomials and the universal Procesi-Schacher conjecture
Positivstellensatz is a fundamental result in real algebraic geometry
providing algebraic certificates for positivity of polynomials on semialgebraic
sets. In this article Positivstellens\"atze for trace polynomials positive on
semialgebraic sets of matrices are provided. A Krivine-Stengle-type
Positivstellensatz is proved characterizing trace polynomials nonnegative on a
general semialgebraic set using weighted sums of hermitian squares with
denominators. The weights in these certificates are obtained from generators of
and traces of hermitian squares. For compact semialgebraic sets
Schm\"udgen- and Putinar-type Positivstellens\"atze are obtained: every trace
polynomial positive on has a sum of hermitian squares decomposition with
weights and without denominators. The methods employed are inspired by
invariant theory, classical real algebraic geometry and functional analysis.
Procesi and Schacher in 1976 developed a theory of orderings and positivity
on central simple algebras with involution and posed a Hilbert's 17th problem
for a universal central simple algebra of degree : is every totally positive
element a sum of hermitian squares? They gave an affirmative answer for .
In this paper a negative answer for is presented. Consequently, including
traces of hermitian squares as weights in the Positivstellens\"atze is
indispensable
Finsler's Lemma for Matrix Polynomials
Finsler's Lemma charactrizes all pairs of symmetric real
matrices and which satisfy the property that for every
nonzero such that . We extend this
characterization to all symmetric matrices of real multivariate polynomials,
but we need an additional assumption that is negative semidefinite outside
some ball. We also give two applications of this result to Noncommutative Real
Algebraic Geometry which for reduce to the usual characterizations of
positive polynomials on varieties and on compact sets.Comment: 23 pages, 2 figures, submitte
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