systems of cubic forms
by
Rainer Dietmann
University of Stuttgart

Wolfgang Schmidt proved in 1982 that any system of r cubic forms in at least 5300r(3r+1)² variables defined over any p-adic field has a common non-trivial p-adic zero. The key to this result were subtle estimates for cubic exponential sums. We show that for p > 2 one can use Hensel's lemma in combination with quadratic exponential sums instead of cubic ones, yielding a simpler proof and a better result needing only about 200r³ variables.

Date received: April 28, 2004