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An algorithm for computing genera of ternary and quaternary quadratic forms. (English) Zbl 0925.11045

Watt, Stephen M. (ed.), ISSAC ’91. Proceedings of the 1991 international symposium on Symbolic and algebraic computation. Bonn, Germany, July 15–17, 1991. New York, NY: ACM Press, 134-143 (1991).
From the introduction: This is a preliminary report on an algorithm for computing genera of ternary and quaternary positive definite quadratic forms over \(\mathbb{Z}\). In Section 1 we collect some basic facts and notations from the theory of quadratic forms and explain the method of neighbouring lattices. The very simple algorithm based on this method is described in Section 2 where one also sees which difficulties arise in higher dimensions. We intend to use our algorithm for the experimental investigation of the Fourier and Fourier-Jacobi coefficients of certain linear combinations of Siegel theta series of quaternary quadratic forms; this is still in progress. The underlying problems from the theory of modular forms which were our starting point for this project are briefly sketched in Section 3. The \(C\)-programs for the implementation of our algorithm are not included, they can be obtained from the author.
For the entire collection see [Zbl 0908.00028].

MSC:

11Y16 Number-theoretic algorithms; complexity
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
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