A counterexample to the weak Shanks conjecture
Complex Variables
2024-05-28 v1 Classical Analysis and ODEs
Functional Analysis
Abstract
We give an example of a function non-vanishing in the closed bidisk and the affine polynomial minimizing the norm of in the Hardy space of the bidisk among all affine polynomials . We show that this polynomial vanishes inside the bidisk. This provides a counterexample to the weakest form of a conjecture due to Shanks that has been open since 1980, with applications that arose from digital filter design. This counterexample has a simple form and follows naturally from [7], where the phenomenon of zeros seeping into the unit disk was already observed for similar minimization problems in one variable.
Cite
@article{arxiv.2405.16943,
title = {A counterexample to the weak Shanks conjecture},
author = {Catherine Bénéteau and Dmitry Khavinson and Daniel Seco},
journal= {arXiv preprint arXiv:2405.16943},
year = {2024}
}