Quintic Forms overp-adic Fields

Abstract

We prove that a quintic form in 26 variables defined over a p-adic field K always has a nontrivial zero over K if the residue class field of K has at least 47 elements. This is in agreement with the theorem of Ax Kochen which states that a homogeneous form of degree d in d 2 +1 variables defined over Q p has a nontrivial Q p -rational zero if p is sufficiently large. The Ax Kochen theorem gives no results on the bound for p. For d=1, 2, 3 it has been known for a long time that there is a nontrivial Q p -rational zero for all values of p. For d=4, Terjanian gave an example of a form in 18 variables over Q 2 having no nontrivial Q 2 -rational zero. This is the first result which gives an effective bound for the case d=5. 1996Academic Press, Inc.

Anisotropic forms modulo p2

“…It has been verified in case d = 2 (see [11] for short proof), in case d = 3 [9,12] and in case d = 5 [11] provided the residue class field has at least 47 elements. But, it is impressive that Ax and Kochen [4], by employing methods from Mathematical Logic, were able to show that Artin's conjecture is very nearly true in general.…”

Zeros of p-adic forms

“…Suppose there exists some a ∈ Z p , such that p 2 | F (a), p F (a) and p | F (a) (17) then there exists an α ∈ Z p such that, F (α) = 0. Moreover, α ≡ a (mod p).…”

Least zero of a cubic form

“…We now apply the work of Leep and Yeomans [13,Lemma 3.3]: if a form of prime degree has a non-trivial zero modulo p and is nondegenerate, then either the form is absolutely irreducible modulo p or it is reducible modulo p. In the first case, we can apply the Lang-Weil estimate in order to deduce that * ( p) = p n−1 + O( p n− 3 2 ), which is satisfactory.…”