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

摘要

Let k be a field of characteristic distinct from 2, d is an element of k*. Let further phi and psi be quadratic forms over k, dim phi = p, dim psi = q. Suppose that the form Phi = phi perpendicular to (t(2) - d) psi is isotropic over the rational function field k(t). We prove that there exists a nontrivial polynomial zero of Phi of degree at most min (2p, 2q, [p+q/i(0) (Phi)] - 1), where i(0) (Phi) is the Witt index of the form Phi, and the degree of a polynomial zero of Phi 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 is an element of k*, forms phi, psi over k, dim phi = p, dim psi = q such that any nontrivial zero of the form Phi = phi perpendicular to (t(2) - d) psi 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. (C) 2015 Elsevier B.V. All rights reserved.