Galois orbits of torsion points over polytopes near atoral sets

TL;DR

This paper proves an equidistribution theorem for torsion points over polytopes near atoral sets in algebraic tori, extending previous results and providing convergence speed estimates and computational algorithms.

Contribution

It generalizes existing equidistribution results to specific polytopes and offers explicit convergence speed estimates and an algorithm for strictness degree calculation.

Findings

Established equidistribution for specific Galois orbits of torsion points.

Provided explicit convergence speed estimates as a negative power of strictness.

Developed an algorithm to compute the strictness degree explicitly.

Abstract

Given an essentially atoral Laurent polynomial $P$, we show an equidistribution theorem for the function $\operatorname{log}|P|$ on specific subsets of Galois orbits of torsion points of the $d$-dimensional algebraic torus $\mathbb{G}^d_m(\overline{\mathbb{Q}})$. The specific subsets under consideration are the preimages of $d$-dimensional polytopes within the hypercube $[0,1]^d$ under the cotropicalization map. This generalises an equidistribution theorem of V. Dimitrov and P. Habegger, who considered only all Galois orbits that correspond to the entire hypercube $[0,1]^d$. In addition, we provide an estimate for the convergence speed of this equidistribution, expressed as a negative power of the strictness degree. Our approach is to derive an alternative version of Koksma's inequality over polytopes. As an application, we provide the convergence speed of heights on a sequence of projective points for a specific two-dimensional example, answering a question posed by R. Gualdi and M. Sombra. In the appendix, we present an algorithm to compute the explicit value of the power of the strictness degree.