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Indefinite quadratic forms in many variables

Published online by Cambridge University Press:  26 February 2010

B. J. Birch  and
H. Davenport
B. J. Birch
Affiliation:
Graduate College, Princeton, N.J.
H. Davenport
Affiliation:
University College, London.
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Let Q(x1, …, xn) be an indefinite quadratic form in n variables with real coefficients. It is conjectured that, provided n ≥ 5, the inequality

is soluble for every ε > 0 in integers x1, …, xn, not all 0. The first progress towards proving this conjecture was made by Davenport in two recent papers; the result obtained involved, however, a condition on the type of the form as well as on n. We say that a non-singular Q is of type (r, n—r) if, when Q is expressed as a sum of squares of n real linear forms with positive and negative signs, there are r positive signs and n—r negative signs. It was proved that (1) is always soluble provided that

Type
Research Article
Information
Mathematika , Volume 5 , Issue 1 , June 1958 , pp. 8 - 12
Copyright
Copyright © University College London 1958

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References

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