Zeros of Laurent multiple orthogonal polynomials on the unit circle

Abstract

We investigate two distinct formulations of Laurent multiple orthogonal polynomials on the unit circle, introduced in arXiv:2410.12094 and arXiv:2601.04783 respectively. For the first formulation, we prove that all zeros lie strictly within the complex open unit disk for any Angelesco or AT system. For the second formulation, we establish normality of all indices of the form $(\mathbf{n};\mathbf{n})$, $(\mathbf{n}+{\mathbf{e}}_j;\mathbf{n})$, and $(\mathbf{n};\mathbf{n}+{\mathbf{e}}_j)$ for any Angelesco or AT system, thereby enabling the full application of the Szegő mapping and Geronimus relations from arXiv:2601.04783 in the multiple orthogonality setting.

14 pages