ABSTRACT
We show that a quadratic form defined over the rational function field ℚ(x 1 , …, x n ) of dimension at least 4.2 n + 1 is isotropic over all fields ℚ p (x 1 , …, x n ), except for finitely many primes. Partial results concerning the u-invariant of p-adic function fields are also shown.
Key Words:
Mathematics Subject Classification:
ACKNOWLEDGMENTS
This paper was written during the author's stay at the Equipe de Logique Mathématique at the Université de Paris 7. During his stay, the author was supported through a Marie Curie Individual Fellowship (Contract HPMF-CT-2001-01384). Part of the work was done while the author was staying at the Department of Mathematics at University College Dublin, during which the author was holding a post-doctoral position in the TMR network “Linear algebraic groups, K-theory and related topics”.
Notes
1A similar statement also follows from a theorem by Greenleaf (Citation1965).
#Communicated by H. Macpherson.