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A determinant of linear forms

Published online by Cambridge University Press:  26 February 2010

H. Davenport
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Extract

Let x1, …, xn be linear forms in u1, …, un with real coefficients, and (for simplicity) determinant 1. Given a form (that is, a homogeneous polynomial) F(x1, …, xn), we can ask the following question: do there exist, for arbitrary real α1, …, αn, integers ul, …, un such that

where C is a suitable number independent of α1, …, αn and of the particular linear forms x1 …, xn? In two well-known cases this is true: namely when

and when

where 1 ≤ r ≤ n − 1.

Type
Research Article
Information
Mathematika , Volume 18 , Issue 1 , June 1971 , pp. 100 - 104
Copyright
Copyright © University College London 1971

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References

1.Davenport, H., Ada Arithmetica, 2 (1937), 262–265.CrossRefGoogle Scholar
2.Hardy, G. H. and Wright, E. M., Theory of Numbers, 4th edition, p. 405. Tchebotareff's proof was published in 1934 but did not become widely known until 1938.Google Scholar
3.Blaney, H., J. London Math. Soc., 23 (1948), 153–160.CrossRefGoogle Scholar
4.Birch, B. J., Acta Arithmetica, 4 (1958), p. 85–98.CrossRefGoogle Scholar
5.Uodewig, E., Matrix Calculus, North-Holland, 1956, p. 57.Google Scholar