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One Cubic Diophantine Inequality
Author(s) -
Freeman D. Eric
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008376
Subject(s) - mathematics , degree (music) , diophantine equation , integer (computer science) , combinatorics , polynomial , homogeneous , homogeneous polynomial , inequality , diophantine approximation , discrete mathematics , mathematical analysis , physics , acoustics , programming language , matrix polynomial , computer science
Suppose that G ( x ) is a form, or homogeneous polynomial, of odd degree d in s variables, with real coefficients. Schmidt [ 15 ] has shown that there exists a positive integer s 0 ( d ), which depends only on the degree d , so that if s ⩾ s 0 ( d ), then there is an x ∈ ℤ s ∖{ 0 } satisfying the inequality | G (x)|<1. (1) In other words, if there are enough variables, in terms of the degree only, then there is a nontrivial solution to (1). Let s 0 ( d ) be the minimum integer with the above property. In the course of proving this important result, Schmidt did not explicitly give upper bounds for s 0 ( d ). His methods do indicate how to do so, although not very efficiently. However, in fact much earlier, Pitman [ 13 ] provided explicit bounds in the case when G is a cubic. We consider a general cubic form F ( x ) with real coefficients, in s variables, and look at the inequality | F (x)|<1. (2) Specifically, Pitman showed that if s ⩾(1314) 256 −1, (3) then inequality (2) is non‐trivially soluble in integers. We present the following improvement of this bound.