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On the Local Solubility of Diophantine Systems

Published online by Cambridge University Press:  04 December 2007

TREVOR D. WOOLEY
TREVOR D. WOOLEY
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109–1003, U.S.A. e-mail: wooley@math.lsa.umich.edu
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Abstract

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Let p be a rational prime number. We refine Brauer‘s elementary diagonalisation argument to show that any system of r homogeneous polynomials of degree d, with rational coefficients, possesses a non-trivial p-adic solution provided only that the number of variables in this system exceeds $(rd^2)^2^(d-1)$. This conclusion improves on earlier results of Leep and Schmidt, and of Schmidt. The methods extend to provide analogous conclusions in field extensions of Q, and in purely imaginary extensions of Q. We also discuss lower bounds for the number of variables required to guarantee local solubility.

Keywords

local solubilitydiophantine equationsforms in many variablesp-adic fields
Type
Research Article
Information
Compositio Mathematica , Volume 111 , Issue 2 , April 1998 , pp. 149 - 165
Copyright
© 1998 Kluwer Academic Publishers