Positive univariate trace polynomials. (English) Zbl 1468.13057
The focus of the paper is to present a sort of Positivstellensatz for univariate trace polynomials. And as a consequence a characterisation of positive univariate trace polynomials restricted to symmetric matrices.
The authors first present some interesting basic results on trace polynomials which are almost immediate from a much older and more abstract result in the 70’s which dealt with multivariate trace polynomials [C. Procesi, Adv. Math. 19, 306–381 (1976; Zbl 0331.15021)]. For instance, we have a nice characterization of when a function between symmetric matrices is actually a univariate trace polynomial (see Prop. 2.1 of the paper). These are not directly used in the main result but are of interest to readers interested in trace polynomials.
The proof of the main result can be found in Section 3 where the authors characterise trace polynomials that are positive when restricted to symmetric matrices in a positivity set defined by a finite set of pure trace polynomials. They also give examples and remarks why some of these conditions are indispensible in this characterization.
The authors first present some interesting basic results on trace polynomials which are almost immediate from a much older and more abstract result in the 70’s which dealt with multivariate trace polynomials [C. Procesi, Adv. Math. 19, 306–381 (1976; Zbl 0331.15021)]. For instance, we have a nice characterization of when a function between symmetric matrices is actually a univariate trace polynomial (see Prop. 2.1 of the paper). These are not directly used in the main result but are of interest to readers interested in trace polynomials.
The proof of the main result can be found in Section 3 where the authors characterise trace polynomials that are positive when restricted to symmetric matrices in a positivity set defined by a finite set of pure trace polynomials. They also give examples and remarks why some of these conditions are indispensible in this characterization.
Reviewer: Jose Capco (Linz)
MSC:
| 13J30 | Real algebra |
| 14P10 | Semialgebraic sets and related spaces |
| 16R30 | Trace rings and invariant theory (associative rings and algebras) |
Citations:
Zbl 0331.15021References:
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