Kleine Nullstellen quadratischer Formen in Funktionenkörpern

Abstract

Letk be a field of characteristic different from 2 andt an indeterminate overk. Letϕ≢0 be a quadratic form inn variables with coefficientsϕ _ij =ϕ _ji ink[t]. We show that if φ vanishes on ad-dimensional subspace ofk(t) ⁿ, then there is a zero (x ₁ ,...,x _n )∈k[t]ⁿ−{(0,...,0)} with max$\{ \deg x_i \} \leqslant \frac{{n - d}}{{2d}}$max{degϕ _ij }. We also show, that the factor$\frac{{n - d}}{{2d}}$ is best possible.