2022/03/01 by Pietro Corvaja, Julian Lawrence Demeio, Julian Demeio +9
Mathematics · #11D75 (Primary) #20F69 (Secondary) #Analytic Number Theory Research #Bounded function #Combinatorics #Discrete mathematics #Double exponential function #Exponential formula #Exponential function #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Mathematical analysis #Mathematics #Multiplicative function #Number Theory (math.NT) #Parametrization (atmospheric modeling) #Physics #Pure mathematics #Simple (philosophy) #math.NT #msc:11D75 #msc:20F69
paper · pdf · doi:10.48550/arxiv.2203.00755
published in arXiv (Cornell University) (Cornell University) · 6 pages; submitted
arxiv created 2022/03/01 · openalex publication_date 2022/03/01 · arxiv updated 2022/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that for a number field F, the distribution of the points of a set Σ⊂ \mathbbAFn with a purely exponential parametrization, for example a set of matrices boundedly generated by semi-simple (diagonalizable) elements, is of at most logarithmic size when ordered by height. As a consequence, one obtains that a linear group Γ⊂ GLn(K) over a field K of characteristic zero admits a purely exponential parametrization if and only if it is finitely generated and the connected component of its Zariski closure is a torus. Our results are obtained via a key inequality about the heights of minimal m-tuples for purely exponential parametrizations. One main ingredient of our proof is Evertse's strengthening of the S-Unit Equation Theorem.