On the number of points over finite fields on varieties related to cluster algebras

TL;DR

This paper calculates the number of finite field points on certain algebraic varieties connected to cluster algebras, specifically focusing on fibers of a projection map from cluster varieties to coefficient space.

Contribution

It provides explicit point counts over finite fields for varieties associated with finite type cluster algebras, advancing understanding of their arithmetic properties.

Findings

Explicit point counts for varieties related to finite type cluster algebras

Identification of fibers of the projection map as key varieties

Enhanced understanding of the arithmetic structure of cluster algebra varieties

Abstract

We compute the number of points over finite fields of some algebraic varieties related to cluster algebras of finite type. More precisely, these varieties are the fibers of the projection map from the cluster variety to the affine space of coefficients.