Voronoi limit measures for iterates of constant-coefficient differential operators on rational functions with simple poles

Abstract

B\o gvad and H\"agg proved that for a rational function with simple poles, the zeros of successive derivatives accumulate on the Voronoi diagram of the pole set, and the normalized zero-counting measures converge to a canonical probability measure supported on this diagram. We extend this result from pure derivatives to iterates of an arbitrary monic constant-coefficient differential operator. Let $h(z)=A(z)/B(z)$ be a reduced rational function, where $B$ is monic of degree $b\ge2$ with distinct zeros $S=\{z_1,\dots,z_b\}$, and let $P(D)=\sum_{j=0}^m c_jD^j$ be a monic constant-coefficient differential operator of order $m\ge1$. After clearing denominators, we can write $P(D)^n(h)=\widetilde A_n/B^{mn+1}$ and study the zeros of the numerator polynomials $\widetilde A_n$. If $r:=\min\{j:c_j\neq0\}$, then (after passing to the proper part of $h$ when $r>0$) the associated zero-counting measures converge vaguely to $$\frac{m(b-1)}{bm-r}\,\mu_S,$$ where $\mu_S$ is the B\o gvad--H\"agg probability measure supported on the Voronoi diagram $V_S$. In particular, the limit is a probability measure exactly when $P(D)=D^m$; otherwise a proportion $\frac{m-r}{bm-r}$ of zeros escapes to infinity (in the sense of vague convergence). When $r<m$, the unshifted logarithmic potentials diverge, but an explicit factorial renormalization yields $L^1_{\mathrm{loc}}(\mathbb C)$ convergence to a subharmonic limit with Riesz measure $\frac{m(b-1)}{bm-r}\,\mu_S$. Apart from this scalar factor, the limiting measure is determined solely by the pole configuration; the coefficients of $P(D)$ affect only an additive constant in the limiting potential.