Kleine Nullstellen homogener quadratischer Gleichungen. (Small zeros of homogeneous quadratic equations)
Let \(f(x_ 1,...,x_ n)=\sum^{n}_{i=1}\sum^{n}_{j=1}f_{ij} x_ ix_ j\) be a quadratic form in \(n\geq 2\) variables with integer coefficients \(f_{ij}=f_{ji}\), not all zero. \textit{J. W. S. Cassels} [Proc. Camb. Philos. Soc. 51, 262-264 (1954; Zbl 0064.283)] proved that if the equation \(f(x_ 1,...,x_ n)=0\) has a solution in integers \(x_ 1,...,x_ n\) not all zero, then it has such a solution satisfying max \(| x_ i| \ll F^{(n-1)/2}\), where the constant in \(\ll\) depends only on n. Here \(F=\max | f_{ij}|.\) Here it is shown that if f vanishes on a d-dimensional rational linear subspace, then there are d linearly independent integer points \({\mathfrak x}_ 1,...,{\mathfrak x}_ d\), such that f vanishes on the subspace generated by \({\mathfrak x}_ 1,...,{\mathfrak x}_ d\) and such that \(| {\mathfrak x}_ 1|...| {\mathfrak x}_ d| \ll F^{(n-d)/2},\) where for \({\mathfrak x}=(x_ 1,...,x_ n)\) we put \(| {\mathfrak x}| =\max | x_ i|.\) This implies in particular that if f is equivalent over the real numbers to a sum of r positive and s negative squares of linear forms with \(r+s=n\), \(r\geq s>0\), \(r\geq s+3\), then f has an integral zero \({\mathfrak x}\) with \(0<| {\mathfrak x}| \ll F^{r/2s}\). This result is best possible as was shown by \textit{W. M. Schmidt} [Trans. Am. Math. Soc. (to appear)].
scientific article; zbMATH DE number 3898966