Quantitative Oppenheim Conjecture for Quadratic Forms in 5 Variables over Function Fields

Abstract

We translate Davenport's and Heilbronn's work on a quantitative version of the Oppenheim conjecture for indefinite diagonal quadratic forms in 5 variables into the setting of function fields.

It has been brought to our attention that our result is a special case of Theorem 2.1 in the following paper: Chih-Nung Zhu, "Diophantine inequalities for the non-Archimedean line F_q((1/T))", Acta Arith. 97, No. 3, 253-267 (2001). We were not aware of this work. Further important work in this direction was done by Craig Spencer