On the size of a linear combination of two linear recurrence sequences over function fields

Abstract

Let G_n and H_m be two non-degenerate linear recurrence sequences defined over a function field F in one variable over mathbb {C}, and let mu be a valuation on F. We prove that under suitable conditions there are effectively computable constants c_1 and C' such that the bound μ(Gn-Hm)≤μ(Gn)+C′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mu (G_n - H_m) \\le \\mu (G_n) + C' \\end{aligned}$$\\end{document}holds for max left( n,m right) > c_1 .