Linear spaces on the intersection of cubic hypersurfaces
Let \(\lambda\) (r,m) denote the least n such that, for any prime p, any set of r cubic forms in n variables over \({\mathbb{Q}}_ p\) has an m- dimensional linear space of common p-adic solutions. Similarly let \(\Lambda\) (r,m) denote the corresponding quantity when \({\mathbb{Q}}_ p\) is replaced by \({\mathbb{Q}}\). The existence of \(\lambda\) (r,m) was established by \textit{R. Brauer} [Bull. Am. Math. Soc. 51, 749-755 (1945)] and that of \(\Lambda\) (r,m) by \textit{B. J. Birch} [Mathematika 4, 102-105 (1957; Zbl 0081.045)]. Indeed in each case a far more general result was obtained. While the works of Brauer and Birch are effective, the estimates for \(\lambda\) (r,m) and \(\Lambda\) (r,m) that one would obtain by following their methods would be extremely large. The present paper gives the bounds \(\lambda (r,m)\ll r^ 2m^ 2+r^ 4m\) for r,\(m\geq 1\), and \(\Lambda (1,m)\ll m^{\alpha}\) for \(m\geq 1\), where \(\alpha ={1\over2}(5+\sqrt{17})=4.56... .\) Moreover it is shown that \(\Lambda (r,m)\ll r^{11}m+r^ 3m^ 5\) and that \(\Lambda (r,m)\ll r^ 5m^{14}\) (r,\(m\geq 1\) in either case). The method is based on elementary algebraic geometry, the only analytic ingredient being an estimate for \(\lambda\) (r,1) due to \textit{W. M. Schmidt} [Monatsh. Math. 93, 211-223 (1982; Zbl 0473.10017)].
existence of m-dimensional linear variety
scientific article; zbMATH DE number 3869428
scientific article