Isotropic subspaces of rational quadratic forms (Q1102994)

Let \(\mathcal F = \mathcal F(x_1,\ldots,x_n)\) be a nondegenerate quadratic form with rational integer coefficients \(f_{ij} = f_{ji}\). Suppose \(\mathcal F\) vanishes on a \(d\)-dimensional rational linear subspace. We give a qualitative and quantitative description of such subspaces. A particular result is as follows: Suppose \(n=2d\). Then there exist linearly independent integral zeros \(\mathfrak x_1,\ldots,\mathfrak x_n\) of \(\mathcal F\) satisfying \[ | \mathfrak x_1| \cdots | \mathfrak x_ n| \ll F^{n/2}, \] where \(F\) is the maximum modulus of the coefficients of \(\mathcal F\) and where for \(\mathfrak x=(x_1,\ldots,x_n)\) we put \(| \mathfrak x| =\max _{1\leq i\leq n}| x_i|\).