Zeros of certain quadratic forms over rational function fields and Prestel's theorem

Abstract

Let k be a field of characteristic distinct from 2, d∈k⁎. Let further φ and ψ be quadratic forms over k, dimφ=p, dimψ=q. Suppose that the form Φ=φ⊥(t2−d)ψ is isotropic over the rational function field k(t). We prove that there exists a nontrivial polynomial zero of Φ of degree at most min⁡(2p,2q,[p+qi0(Φ)]−1), where i0(Φ) is the Witt index of the form Φ, and the degree of a polynomial zero of Φ is understood as the largest degree of its components. Also we show that for any positive integers p and q there exists a field k, d∈k⁎, forms φ, ψ over k, dimφ=p, dimψ=q such that any nontrivial zero of the form Φ=φ⊥(t2−d)ψ has degree at least min⁡(p+1,q). In particular, we show that the upper bound on the degrees of zeros of forms in Prestel's theorem [6] is at most two times bigger than the strict bound.