J. Aust. Math. Soc.
82 (2007), 221236

On systems of diagonal forms

Michael P. Knapp
Mathematical Sciences Department
Loyola College
4501 North Charles Street
Baltimore, MD 212102699
USA
mpknapp@loyola.edu



Abstract

In this paper we consider systems of diagonal
forms with integer coefficients in which each
form has a different degree. Every such system
has a nontrivial zero in every padic field
provided that the number of variables is
sufficiently large in terms of the degrees.
While the number of variables required grows at
least exponentially as the degrees and number of
forms increase, it is known that if
p
is sufficiently large then only a small
polynomial bound is required to ensure zeros in
. In this paper we explore the question of how
small we can make the prime
p
and still have a polynomial bound. In
particular, we show that we may allow
p
to be smaller than the largest of the degrees.

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