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Journal of the London Mathematical Society 2000 61(1):25-35; doi:10.1112/S0024610799008376
© 2000 by London Mathematical Society
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© The London Mathematical Society

One Cubic Diophantine Inequality

D. Eric Freeman

Department of Mathematics, University of Michigan East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, USA

Received 15 May 1998.

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 s0(d), which depends only on the degree d, so that if s ≥ s0(d), then there is an x isin Zs\{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 s0(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 s0(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.


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