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15 August 2004 Local densities and explicit bounds for representability by a quadratic form
Jonathan Hanke
Duke Math. J. 124(2): 351-388 (15 August 2004). DOI: 10.1215/S0012-7094-04-12424-8


In this paper we give explicit lower bounds for an integer m to be represented by a positive definite integral quadratic form Q in n≥3 variables defined over ℚ. As an example, we apply these bounds to answer affirmatively the long-standing conjecture of Kneser that the only positive integers not represented by x2+3y2+5z2+7w2 are 2 and 22.

When n=3, the existence of spinor square classes and the possible existence of a Siegel zero complicates the estimate and requires us to restrict m to a finite union of square classes in order to obtain explicit constants. In this setting, we obtain a lower bound and asymptotics for the number of representations of m by Q, even within a spinor square class.

These methods can be easily generalized to obtain similar results for the representability of integers by a totally definite quadratic form over a totally real number field, and we carry out our local analysis in this generality. We also describe how to generalize these results to handle congruence conditions and representability by a rational quadratic polynomial.


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Jonathan Hanke. "Local densities and explicit bounds for representability by a quadratic form." Duke Math. J. 124 (2) 351 - 388, 15 August 2004.


Published: 15 August 2004
First available in Project Euclid: 5 August 2004

zbMATH: 1090.11023
MathSciNet: MR2079252
Digital Object Identifier: 10.1215/S0012-7094-04-12424-8

Primary: 11D09
Secondary: 11E12 , 11E20 , 11E25 , 11Y50

Rights: Copyright © 2004 Duke University Press


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Vol.124 • No. 2 • 15 August 2004
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