#### Vol. 1, No. 1, 2013

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Solving quadratic equations in dimension $5$ or more without factoring

### Pierre Castel

Vol. 1 (2013), No. 1, 213–233
##### Abstract

Let $Q$ be a $\mathfrak{5}×\mathfrak{5}$ symmetric matrix with integral entries and with $detQ\ne \mathfrak{0}$, but neither positive nor negative definite. We describe a probabilistic algorithm which solves the equation ${\phantom{\rule{1em}{0ex}}}^{t}\phantom{\rule{0.3em}{0ex}}xQx=\mathfrak{0}$ over $ℤ$ without factoring $detQ$. The method can easily be generalized to forms of higher dimensions by reduction to a suitable subspace.

##### Keywords
general quadratic forms, factorization, general quadratic equations, isotropic spaces, algorithmic number theory, Cebotarev density theorem
Primary: 11E20
Secondary: 11D09
##### Milestones
Published: 14 November 2013
##### Authors
 Pierre Castel Laboratoire de Mathématiques Nicolas Oresme Université de Caen Basse-Normandie UMR CNRS 6139 14032 Caen France