Small zeros of quadratic forms over rational function fields (Q1361000)

Let \(\varphi (X_1,\ldots ,X_n)=\sum_{1\leq i,j\leq n} \varphi_{ij}X_iX_j\) be a non-zero quadratic form with coefficients in \({\mathbb{Z}}\). Suppose that \(\varphi_{ij}=\varphi_{ji}\). Let \(\Phi := \max_{1\leq i,j\leq n} | \varphi_{ij}| \). In 1955, Cassels showed that if the equation \(\varphi ({\mathbf x})=0\) has a nontrivial solution \({\mathbf x}\in{\mathbb{Z}}^n\) then it has such a solution with \(| {\mathbf x}| \ll \Phi^{(n-1)/2}\), where \(| {\mathbf x}| \) denotes the maximum of the absolute values of the coordinates of \({\mathbf x}\). In 1985, \textit{H. P. Schlickewei} [Monatsh. Math. 100, 35-45 (1985; Zbl 0564.10012)] generalised this as follows: if \(d\) is the largest integer with the property that there exists a \(d\)-dimensional linear subspace \(V\) of \({\mathbb{Q}}^n\) on which \(\varphi\) vanishes identically, then there are linearly independent vectors \({\mathbf x}_1,\ldots ,{\mathbf x}_d\in V\cap {\mathbb{Z}}^n\) with \(| {\mathbf x}_1| \cdots | {\mathbf x}_d| \ll | \Phi | ^{(n-d)/2}\). In the present paper, the author proves a function field analogue of this result. More precisely, he proves the same result as Schlickewei's, but with \({\mathbb{Z}}\) being replaced by the polynomial ring \(k[t]\) where \(k\) is a field of characteristic \(\not= 2\) and with the usual absolute value on \({\mathbb{Z}}\) being replaced by \(| f| =e^{\text{deg }f}\) for \(f\in k[t]\). Like Schlickewei, the author uses geometry of numbers. In particular, he uses Mahler's function field analogue of Minkowski's theorem.