The Stieltjes–Fekete Problem and Degenerate Orthogonal Polynomials

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 energy with logarithmic interactions under an external field. The optimal configuration satisfies an algebraic set of equations: we call this set of algebraic equations the Stieltjes–Fekete problem. In this work we consider the Stieltjes-Fekete problem when the derivative of the external field is an arbitrary rational complex function. We show that, under assumption of genericity, its solutions are in one-to-one correspondence with the zeroes of certain non-hermitian orthogonal polynomials that satisfy an excess of orthogonality conditions and are thus termed “degenerate”. When the differential of the external field on the Riemann sphere is of degree our result reproduces Stieltjes’ original result and provides its direct generalization for higher degree after more than a century since the original result.