Positive univariate trace polynomials

Igor Klep; James Eldred Pascoe; Jurij Volčič

doi:10.1016/j.jalgebra.2021.03.027

Positive univariate trace polynomials

Igor Klep, James Eldred Pascoe, Jurij Volčič

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(x^j). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.

Original language	English
Pages (from-to)	303-317
Number of pages	15
Journal	Journal of Algebra
Volume	579
DOIs	https://doi.org/10.1016/j.jalgebra.2021.03.027
State	Published - 1 Aug 2021
Externally published	Yes

Keywords

Hankel matrix
Positivstellensatz
Power mean inequality
Real algebraic geometry
Trace polynomial

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1016/j.jalgebra.2021.03.027

Cite this

@article{91492870b3b34f198f52be3d9cc4fcf8,

title = "Positive univariate trace polynomials",

abstract = "A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(xj). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.",

keywords = "Hankel matrix, Positivstellensatz, Power mean inequality, Real algebraic geometry, Trace polynomial",

author = "Igor Klep and Pascoe, {James Eldred} and Jurij Vol{\v c}i{\v c}",

note = "Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

year = "2021",

month = aug,

day = "1",

doi = "10.1016/j.jalgebra.2021.03.027",

language = "English",

volume = "579",

pages = "303--317",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Positive univariate trace polynomials

AU - Klep, Igor

AU - Pascoe, James Eldred

AU - Volčič, Jurij

PY - 2021/8/1

Y1 - 2021/8/1

N2 - A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(xj). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.

AB - A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(xj). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.

KW - Hankel matrix

KW - Positivstellensatz

KW - Power mean inequality

KW - Real algebraic geometry

KW - Trace polynomial

UR - http://www.scopus.com/inward/record.url?scp=85103698124&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2021.03.027

DO - 10.1016/j.jalgebra.2021.03.027

M3 - Article

AN - SCOPUS:85103698124

SN - 0021-8693

VL - 579

SP - 303

EP - 317

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Positive univariate trace polynomials

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this