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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Solving quadratic equations using reduced unimodular quadratic forms
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by Denis Simon PDF
Math. Comp. 74 (2005), 1531-1543 Request permission


Let $Q$ be an $n\times n$ symmetric matrix with integral entries and with $\det Q \neq 0$, but not necesarily positive definite. We describe a generalized LLL algorithm to reduce this quadratic form. This algorithm either reduces the quadratic form or stops with some isotropic vector. It is proved to run in polynomial time. We also describe an algorithm for the minimization of a ternary quadratic form: when a quadratic equation $q(x,y,z)=0$ is solvable over $\mathbb {Q}$, a solution can be deduced from another quadratic equation of determinant $\pm 1$. The combination of these algorithms allows us to solve efficiently any general ternary quadratic equation over $\mathbb {Q}$, and this gives a polynomial time algorithm (as soon as the factorization of the determinant of $Q$ is known).
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Additional Information
  • Denis Simon
  • Affiliation: LMNO–UMR 6139, Université de Caen–Campus II, Bd du Maréchal Juin, BP 5186–14032 Caen Cedex, France
  • Email:
  • Received by editor(s): February 14, 2003
  • Received by editor(s) in revised form: February 26, 2004
  • Published electronically: January 27, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 1531-1543
  • MSC (2000): Primary 11Y50, 11E20; Secondary 11H55
  • DOI:
  • MathSciNet review: 2137016