Asymptotic zero distribution conjecture for Gonchar polynomials of the third kind

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Let Γ\Gamma be the set consisting of the boundary of the union of the two unit disks centered at −1-1 and 00, together with the line segment connecting their intersection points. Let G3⁡(d,q′;z)\operatorname{G_3}(d,q^\prime;z) denote the Gonchar polynomial of the third kind in dimension dd with conical charge q′q^\prime. Asymptotic zero distribution conjecture. As d→∞d\to\infty, all zeros of G3⁡(d,q′;z)\operatorname{G_3}(d,q^\prime;z) tend to Γ\Gamma, and every point of Γ\Gamma attracts zeros of these polynomials. This conjectural description identifies the limiting zero set for the Gonchar polynomials of the third kind; the supplied text does not indicate whether it has been proved or remains open.

References

Primary source

Johann S. Brauchart, Peter D. Dragnev and Edward B. Saff, “An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge”, arXiv:1402.3367 (2014).

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