A counterexample to the weak Shanks conjecture

TL;DR

This paper presents a counterexample to the weak Shanks conjecture, demonstrating that an affine polynomial minimizing a Hardy space norm can have zeros inside the bidisk, challenging previous assumptions since 1980.

Contribution

It provides the first explicit counterexample to the weakest form of Shanks' conjecture, revealing zeros inside the bidisk for a specific minimization problem.

Findings

Counterexample polynomial vanishes inside the bidisk

Challenges the weak Shanks conjecture

Zeros can seep into the unit disk in minimization problems

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.