Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.
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.
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