Stable symmetric polynomials and the Schur-Agler class (Q358944)

A multivariable polynomial \(p\in \mathbb{C}[z_1,\ldots,z_n]\) is stable if \(p\) has no zeros on the closed polydisk \(\overline{\mathbb{D}}^n\). One variable stable polynomials \(p\in \mathbb{C}[z]\) satisfy a Christoffel-Darboux formula as well as the classical Grace-Walsh-Szegő theorem related with the stability of its multi-affine symmetrization (multi-affine means a multivariable polynomial of degree at most one in each variable separately). To generalize the Christoffel-Darboux formula to multivariable polynomials is an interesting question. It is known for two variables, however for three or more variables it holds for Agler denominators, being denominators of rational inner functions in the Schur-Agler class, within the bounded analytic functions on the polydisk.NEWLINENEWLINEIn the paper under review, necessary and sufficient conditions for a multi-affine symmetric polynomial to be an Angler denominator are given. For three variables it was proved in [\textit{A. Kummert}, ``Synthesis of 3-D lossless first-order one ports with lumped elements'', IEEE Trans. Circuit Systems 36, 1445--1449 (1989; \url{doi:10.1109/31.41302})] that all multi-affine stable polynomials are Agler denominators. In this paper, Kummert's result is sharpened.