Asymptotic zero distribution conjecture for Gonchar polynomials of the third kind

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 qq^\prime. Asymptotic zero distribution conjecture. As dd\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.

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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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