Forms in many variables over p-adic fields.

Abstract

In this dissertation, we prove several results about the number of variables necessary to ensure that forms (homogeneous polynomials) with coefficients in a p -adic field <blkbd>K</blkbd> have nontrivial zeros. First, we obtain a bound on the size of the residue field of <blkbd>K</blkbd> needed to ensure that a form of degree <italic>k</italic> = 7 or 11 in <italic>k</italic><super>2</super> + 1 variables has a nontrivial zero in <blkbd>K</blkbd> . We also show that a system of <italic>R</italic> additive forms of degree <italic>k</italic> in <italic>s</italic> variables will have a nontrivial zero over <blkbd>Qp</blkbd> provided that <italic>s</italic> &ge; 4<italic>R</italic><super>2 </super><italic>k</italic><super>2</super>, and prove a similar result when <blkbd>K</blkbd> is a finite extension of <blkbd>Qp</blkbd> . If we have <italic>k</italic> = <italic>p</italic><super>tau</super>, then we obtain another similar bound which does not depend on the degree of the field extension. Finally, we obtain an upper bound on the smallest number of variables needed to ensure that a system of two additive forms of degrees <italic>k</italic> and <italic>n</italic> = <italic>k</italic> - <italic> q</italic><super>tau</super>, where <italic>q</italic> is an odd prime and tau is a nonnegative integer, has a nontrivial solution over <blkbd>Qp</blkbd> for every <italic>p</italic>.