We give an example of a function f non-vanishing in the closed bidisk together with the affine polynomial minimizing the norm of \(1-pf\) in the Hardy space of the bidisk among all affine polynomials p and show that this polynomial vanishes inside the bidisk. This function f has a simple form and follows naturally from Bénéteau (Rev Mat Iberoam 35(2):607–642, 2019), where the phenomenon of zeros seeping into the unit disk was already observed for similar minimization problems in one variable. We can then deduce 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.
