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Asymptotic formulae for pairs of diagonal equations

Published online by Cambridge University Press:  07 July 2004

JÖRG BRÜDERN
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D-70511 Stuttgart, Germany. e-mail: [email protected]
TREVOR D. WOOLEY
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Ave., Ann Arbor, MI 48109-1109, U.S.A. e-mail: [email protected]

Abstract

Consider a system of diagonal equations \begin{equation}\sum_{j=1}^sa_{ij}x_j^k=0\quad (1\le i\le r),\end{equation} satisfying the property that the (fixed) integral coefficient matrix $(a_{ij})$ contains no singular $r\times r$ submatrix. A recent paper of the authors [3] establishes that whenever $k\ge 3$ and $s>(3r+1)2^{k-2}$, then the expected asymptotic formula holds for the number $N(P)$ of integral solutions ${\bf x}$ of ($1{\cdot}1$) with $|x_i|\le P$$(1\le i\le s)$.

Type
Research Article
Copyright
2004 Cambridge Philosophical Society

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