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A counterexample to the weak Shanks conjecture
Published 27 May 2024 in math.CV, math.CA, and math.FA | (2405.16943v1)
Abstract: We give an example of a function 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 . 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.
- P. Delsarte, Y.V. Genin, and Y.G. Kamp, Comments on “proof of a modified form of Shanks’ conjecture on the stability of 2-D planar least square inverse polynomials and its implications”, IEEE Trans. Circuits Syst. CAS-32 (1985) no. 9, 966–968.
- P. Karivaratharajan and H. Reddy, A Note on LSI polynomials and Comments on a “Proof of a Modified form of Shanks’ Conjecture”, IEEE Transactions on Circuits and Systems 32 (1985) no. 9, 966-968.
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