2020/05/15 by Paloma Bengoechea · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Arithmetic #Binary number #Combinatorics #Degree (music) #Discrete mathematics #Discriminant #Integer (computer science) #Mathematical analysis #Mathematics #Physics #Rational number #Statistics #Upper and lower bounds #Value (mathematics) #math.NT #semigroups and automata theory
paper · pdf · doi:10.1093/imrn/rnaa136
published in International Mathematics Research Notices 2022(2), 1217-1244 (Oxford University Press) · arXiv admin note: text overlap with arXiv:1906.03705
openalex publication_date 2020/05/15 · arxiv created 2020/08/22 · arxiv updated 2020/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract Let F(x, y) be a binary form with integer coefficients, degree n≥ 3, and irreducible over the rationals. Suppose that only s + 1 of the n + 1 coefficients of F are nonzero. We show that the Thue inequality |F(x,y)|≤ m has ≪ s m2/n solutions provided that the absolute value of the discriminant D(F) of F is large enough. We also give a new upper bound for the number of solutions of |F(x,y)|≤ m, with no restriction on the discriminant of F that depends mainly on s and m, and slightly on n. Our bound becomes independent of m when m<|D(F)|2/(5(n-1)), and also independent of n if |D(F)| is large enough.