M/E 23

In 1959 Davenport, Birch and Lewis proved that any cubic form over the field of p-adic numbers in more than nine variables has a nontrivial zero. One of the main steps in their proof is the observation that it is sufficient to prove this statement for a Zariski dense subset of the space of all cubic forms, defined by an invariant. They later used the same technique to show that systems of three quadratic forms over a local number field with large enough residue field in more than twelve variables have a nontrivial zero. I show that this method of reduction can be generalized to systems of arbitrary forms over a henselian discretely valued field of characteristic zero. This result is relevant for new progress on systems of quadratic forms over complete discretely valued fields made by David Leep.

Sten Veraa

Universiteit Antwerpen