Work

Matrix Fejér-Riesz type theorem for a union of an interval and a point

Year: 2026

Type: article

Abstract: The matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In [31] this was extended to the characterization on arbitrary closed semialgebraic sets K ⊆ R by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when K is the whole line, an unbounded interval, a union of two unbounded intervals, and according to a conjecture of [31] also when K is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem on a union of a bounded interval and a point. The presented technique for solving the corresponding moment problem can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets K [31, Theorem C] .

Source: Journal of Pure and Applied Algebra

DOI: 10.1016/j.jpaa.2026.108173

Authors: Shengding Sun, Aljaž Zalar

Institutions: University of Cambridge, University of Ljubljana, Institute of Mathematics, Physics, and Mechanics

Language: English

Topic: Mathematical functions and polynomials

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funders: Javna Agencija za Raziskovalno Dejavnost RS, HORIZON EUROPE Framework Programme