Diophantine inequality involving binary forms

• Published in: Frontiers of mathematics in China Vol. 12; no. 6; pp. 1457 - 1468
• Main Author: MU, Quanwu
• Format: Journal Article
• Language: English
• Published: Beijing Higher Education Press 01.12.2017; Springer Nature B.V
• Subjects: binary form; Davenport-Heilbronn method; Diophantine inequality; Integers; Mathematical analysis; Mathematics; Mathematics and Statistics; Real numbers; Research Article; 11P55; Davenport–Heilbronn method; binary form; 11D75; Diophantine inequality
• ISSN: 1673-3452; 1673-3576
• DOI: 10.1007/s11464-017-0602-y

Let d ≥ 3 be an integer, and set r = 2^d-1 ＋ 1 for 3 ≤ d ≤ 4, r = 17 5~ ＂2441 for 5 ≤ d ≤ 6, r = d^2＋d＋1 for 7 ≤ d ≤ 8, and r = d^2＋d＋2 for d ≥ 9, respectively. Suppose that Фi（x, y） E Z[x, y] （1 ≤ i ≤ r） are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2,. ..., λr are nonzero real numbers with λ1/λ2 irrational, and λ1λ1（x1, y1） ＋ λ2q）2（x2, y2） ＋ ... ＋ ）,λrФr（xr, yr） is indefinite. Then for any given real η and σ with 0 〈 cr 〈 22-d, it is proved that the inequalityhas infinitely ｜r∑i=1λФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-σmany solutions in integers Xl, x2,..., xr, Yl, Y2,.--, Yr. This result constitutes an improvement upon that of B. Q. Xue.