J. Aust. Math. Soc.  82 (2007), 221-236
On systems of diagonal forms

Michael P. Knapp
  Mathematical Sciences Department
  Loyola College
  4501 North Charles Street
  Baltimore, MD 21210-2699
  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 p-adic field $\mathbb{Q}_p$ 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 $\mathbb{Q}_p$. 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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