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Conditions for the solvability of systems of two and three additive forms over p-adic fields

Published online by Cambridge University Press:  19 October 2005

H. Godinho
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, DF 70910-900, Brazil. E-mail: hemar@mat.unb.br
P. H. A. Rodrigues
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO 74001-970, Brazil. E-mail: paulo@mat.ufg.br
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Abstract

This paper is concerned with non-trivial solvability in $p$-adic integers of systems of two and three additive forms. Assuming that the congruence equation $a x^k + b y^k + c z^k \equiv d \,(\mbox{mod}\,p)$ has a solution with $xyz \not\equiv 0\,(\mbox{mod}\,p)$ we have proved that any system of two additive forms of odd degree $k$ with at least $6 k + 1$ variables, and any system of three additive forms of odd degree $k$ with at least $14 k + 1$ variables always has non-trivial $p$-adic solutions, provided $p$ does not divide $k$. The assumption of the solubility of the congruence equation above is guaranteed for example if $p > k^4$.

In the particular case of degree $k = 5$ we have proved the following results. Any system of two additive forms with at least $n$ variables always has non-trivial $p$-adic solutions provided $n \geq 31$ and $p > 101$ or $n \geq 36$ and $p > 11$. Furthermore any system of three additive forms with at least $n$ variables always has non-trivial $p$-adic solutions provided $n \geq 61$ and $p > 101$ or $n \geq 71$ and $p > 11$.

Type
Research Article
Copyright
2005 London Mathematical Society

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Footnotes

The research of the first author was partially supported by a grant from CNPq-Brasil, and that of the second author was partially supported by a grant from CAPES-Brasil.

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