Equidistribution of the conjugates of algebraic units

TL;DR

The paper develops a two-tier approach to equidistribution of zeros: first for univariate polynomials via height relative to a capacity $\text{cap}_{0,\infty}$, then for bivariate polynomial mappings through homogeneous potential theory. Central to the method is lifting a univariate polynomial to a homogeneous map in $\mathbb{C}^2$ and proving a new equidistribution result for zeros of such maps when the homogeneous height ${\mathscr H}_{\Sigma}(F)$ tends to zero and $\Sigma$ has ${\rm cap}_h(\Sigma)=1$. The main results include Theorem ${\['T:units_equi'\]}$, which gives equidistribution of conjugates of algebraic units on a set $K$ with ${\rm cap_{0, \infty}}(K)=1$, and Theorem ${\['T:homo_equi'\]}$, a general equidistribution statement for zeros of generic polynomial mappings in $\mathbb{C}^2$. These results extend prior work of Bilu and Rumely to broader Cantor-capacity-1 sets, provide new arithmetic probability measures, and offer a robust framework for constructing equidistribution phenomena via 2D polynomial lifts. The appendix generalizes the constructions to nonpolar and nonregular scenarios, ensuring broad applicability of the theory.

Abstract

We prove an equidistribution result for the zeros of polynomials with integer coefficients and simple zeros. Specifically, we show that the normalized zero measures associated with a sequence of such polynomials, having small height relative to a certain compact set in the complex plane, converge to a canonical measure on the set. In particular, this result gives an equidistribution result for the conjugates of algebraic units, in the spirit of Bilu's work. Our approach involves lifting these polynomials to polynomial mappings in two variables and proving an equidistribution result for the normalized zero measures in this setting.

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.4
• Remark 2.1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof