We prove that a quintic form in 26 variables defined over a p-adic field K always has a non trivial 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 d2 + 1 variables defined over Qp has a nontrivial Qp-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 Qp-rational zero for all values of p. For d=4, Terjanian gave an example of a form in 18 variables over Q2 having no nontrivial Q2-rational zero. This is the first result which gives an effective bound for the case d= 5.
|Number of pages||11|
|Journal||Journal of Number Theory|
|State||Published - Apr 1996|
ASJC Scopus subject areas
- Algebra and Number Theory