Duke Mathematical Journal

Local densities and explicit bounds for representability by a quadratic form

Jonathan Hanke

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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.

Article information

Duke Math. J. Volume 124, Number 2 (2004), 351-388.

First available in Project Euclid: 5 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D09: Quadratic and bilinear equations
Secondary: 11E25: Sums of squares and representations by other particular quadratic forms 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11Y50: Computer solution of Diophantine equations 11E12: Quadratic forms over global rings and fields


Hanke, Jonathan. Local densities and explicit bounds for representability by a quadratic form. Duke Math. J. 124 (2004), no. 2, 351--388. doi:10.1215/S0012-7094-04-12424-8. https://projecteuclid.org/euclid.dmj/1091735978

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