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Effective Hasse principle for the intersection of two quadrics

Part of: Diophantine equations Computational number theory

Published online by Cambridge University Press:  26 August 2016

Tony Quertier
Tony Quertier*
Affiliation:
UMR 6139 - Laboratoire de Mathématiques Nicolas Oresme (LMNO), Université de Caen Normandie UFR des Sciences, Campus 2, Côte de nacre Bd Maréchal Juin, 14032 Caen cedex 5, France email tony.quertier@unicaen.fr

Abstract

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We consider a smooth system of two homogeneous quadratic equations over $\mathbb{Q}$ in $n\geqslant 13$ variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over $\mathbb{R}$ . In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists and, if so, compute one.

MSC classification

Primary: 11D09: Quadratic and bilinear equations 11Y50: Computer solution of Diophantine equations

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