The Stieltjes-Fekete problem and degenerate orthogonal polynomials
Tamara Grava (University of Bristol and SISSA)
Abstract: A result of Stieltjes famously relates the zeroes of the classical orthogonal polynomials with the configurations of points on the line that minimize a suitable logarithmic energy, or equivalently the solutions of a suitable weighted Fekete problem. The optimal configuration satisfies an algebraic set of equations with the logarithmic derivative of the weight function as ``external field": we call this set of algebraic equations the Stieltjes-Fekete problem. In this work we consider the Stieltjes-Fekete problem with an arbitrary rational external field. We show that its solutions are in one-to-one correspondence with the zeroes of certain non-hermitean orthogonal polynomials that satisfy an excess of orthogonality conditions and are thus termed ``degenerate". This generalizes the above mentioned result of Stieltjes.
complex variablesdynamical systems
Audience: researchers in the topic
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