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Abstract:Let $Q$ be a monic polynomial of degree $M+1$. We study the algebraic system \[
\sum_{j\ne i}\frac{1}{x_i-x_j}=Q(x_i),\qquad i=1,\ldots,N, \] for pairwise distinct complex numbers $x_1,\ldots,x_N$, modulo permutations of these numbers. The case $M=0$ is, after a translation, the classical Stieltjes system for the zeros of a Hermite polynomial. We prove that, for arbitrary $Q$, the number of solutions is at most $\binom{N+M}{N}$, and that the coefficient equations for the associated monic Stieltjes polynomial have total intersection multiplicity exactly $\binom{N+M}{N}$. Consequently the bound is attained for all $Q$ in a non-empty Zariski open subset of the affine space of monic polynomials of degree $M+1$. We also describe the solutions when the coefficient of the linear term of $Q$ is large: the system splits into $M+1$ weakly coupled classical Stieltjes systems, one near each zero of $Q$.

Submission history

From: Boris Shapiro [view email]
[v1] Sun, 14 Jun 2026 16:45:36 UTC (8 KB)