$p$-adic zeros of quintic forms

Abstract

It is shown that a quintic form over a p-adic field with at least 26 variables has a non-trivial zero, providing that the cardinality of the residue class field exceeds 9.

Two Cases of Artin's Conjecture

“…, k R ) (see Brown [5] and Cohen [11]), these bounds contain nested exponentials and are, therefore, huge. For some small values of k there are better bounds for p 0 (k) known, for example, p 0 (5) ≤ 7 by Dumke [18] and both p 0 (7) ≤ 883 and p 0 (11) ≤ 8053 by Wooley [43].…”