Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415796 | Journal of Pure and Applied Algebra | 2016 | 11 Pages |
Let k be a field of characteristic distinct from 2, dâkâ. Let further Ï and Ï be quadratic forms over k, dimÏ=p, dimÏ=q. Suppose that the form Φ=Ïâ¥(t2âd)Ï is isotropic over the rational function field k(t). We prove that there exists a nontrivial polynomial zero of Φ of degree at most minâ¡(2p,2q,[p+qi0(Φ)]â1), 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, dâkâ, forms Ï, Ï over k, dimÏ=p, dimÏ=q such that any nontrivial zero of the form Φ=Ïâ¥(t2âd)Ï has degree at least minâ¡(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.