2024/12/15 by C.-J. Lin, Chenying Lin, Lin, Chenying · 1 citation
Mathematics · #11G50 #37P30 #52B11 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11J83 #Secondary 14G40
paper · doi:10.48550/arxiv.2412.11156
Given an essentially atoral Laurent polynomial P, we show an equidistribution theorem for the function log|P| on specific subsets of Galois orbits of torsion points of the d-dimensional algebraic torus \mathbbGdm(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.