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ONE CUBIC DIOPHANTINE INEQUALITY

Published online by Cambridge University Press:  01 February 2000

D. ERIC FREEMAN
D. ERIC FREEMAN
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, USA
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Abstract

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 [ges ] s0(d), then there is an x ∈ ℤs\{0} satisfying the inequality

formula here

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

formula here

Specifically, Pitman showed that if

formula here

then inequality (2) is non-trivially soluble in integers. We present the following improvement of this bound.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 61 , Issue 1 , February 2000 , pp. 25 - 35
Copyright
The London Mathematical Society 2000

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