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arXiv · 2405.16943

A counterexample to the weak Shanks conjecture

Catherine Bénéteau·Dmitry Khavinson·Daniel Seco
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Abstract

We give an example of a function $f$ non-vanishing in the closed bidisk and the affine polynomial minimizing the norm of $1-pf$ in the Hardy space of the bidisk among all affine polynomials $p$. 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.

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Catherine Bénéteau, Dmitry Khavinson, Daniel Seco. 2024-05-27. A counterexample to the weak Shanks conjecture. https://arxiv.org/abs/2405.16943

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