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Indefinite quadratic forms in many variables
Published online by Cambridge University Press: 26 February 2010
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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
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References
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