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Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc

  • Oliver Navío
  • Daniel Seco
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Abstract

We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.

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Cite

BibTeXRIS
BibTeX
@misc{navio2026polynomial,
  title = {{Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc}},
  author = {Oliver Navío and Daniel Seco},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37180},
  eprint = {2609.37180},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37180},
}
RIS
TY  - UNPB
TI  - Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc
AU  - Oliver Navío
AU  - Daniel Seco
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37180
UR  - https://arxiv.org/abs/2609.37180
AB  - We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.
ER  -