Positivstellensätze for noncommutative rational expressions

Pascoe, J.

doi:10.1090/proc/13773

Positivstellensätze for noncommutative rational expressions
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by J. E. Pascoe PDF

Proc. Amer. Math. Soc. 146 (2018), 933-937 Request permission

Abstract:

We derive some Positivstellensätze for noncommutative rational expressions from the Positivstellensätze for noncommutative polynomials. Specifically, we show that if a noncommutative rational expression is positive on a polynomially convex set, then there is an algebraic certificate witnessing that fact. As in the case of noncommutative polynomials, our results are nicer when we additionally assume positivity on a convex set, that is, we obtain a so-called “perfect Positivstellensatz” on convex sets.

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