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Weak Shanks conjecture

Let D2={z∈C2:∣z1∣<1, ∣z2∣<1}\mathbb{D}^2=\{z\in\mathbb{C}^2:|z_1|<1,\ |z_2|<1\}, let α>0\alpha>0, and let Dα(D2)\mathcal{D}_\alpha(\mathbb{D}^2) be the corresponding Dirichlet-type space. For every polynomial ff satisfying f(z)≠0f(z)\neq 0 for all z∈D‾2z\in\overline{\mathbb{D}}^2, let pfp_f be the affine polynomial minimizing ∥pf−1∥Dα(D2)\|pf-1\|_{\mathcal{D}_\alpha(\mathbb{D}^2)} over all affine polynomials pp. The Weak Shanks conjecture asserts that pf(z)≠0p_f(z)\neq 0 for every z∈D2z\in\mathbb{D}^2.

References

Primary source

arXiv

Additional references

approximation-theorycomplex-analysisfunction-spaces
Posted · edited

Progress summary

Refreshed
Claimed solved

A 2024 bidisc counterexample disproved the conjecture, and a 2026 preprint claims the failure extends to every positive parameter and higher dimensions.

The Weak Shanks conjecture predicts that optimal polynomial approximants to a zero-free function remain zero-free. Béneteau, Khavinson, and Seco reported a counterexample in the Hardy space of the bidisc, disproving its weakest form, which had been open since 1980.

Known results

  • Genin and Kamp disproved the original stronger conjecture for polynomial denominators.
  • Delsarte, Genin, and Kamp showed that an attempted proof of the weaker conjecture failed.
  • Béneteau, Khavinson, and Seco (2024) produced a Hardy-bidisc counterexample and extended it to higher-dimensional polydiscs.
  • Their Dirichlet-type extension was established only for sufficiently small positive α\alpha, leaving the full positive range open.

September 2026 parameter-range extension

Navío and Seco’s preprint claims counterexamples for every positive Dirichlet-type parameter and further anisotropic and higher-dimensional examples. This strengthens the known result, but remains an unverified preprint claim.

Current status (as of September 2026): The Hardy-bidisc conjecture is settled by claimed counterexamples, while the all-α>0\alpha>0 extension and several higher-dimensional questions remain unverified or open.

Sources

Solutions 0

No solutions have been posted yet.