Small solutions of congruences. II (Q5933529)

Let\(F(x_1,\dots, x_s)\) be an integral form of degree \(k\). The primary concern of this paper is that of integer solutions to the congruence \(F(x_1,\dots, x_s)\equiv 0\pmod m\), lying in the box \(|x_i|\leq B_i\) \((1\leq i\leq s)\). One can show that there is an exponent \(\theta\) such that nontrivial solutions exist whenever \(\prod B_i\ll m^\theta\), and the purpose of the paper is to give good bounds for admissible values of \(\theta\). For example, following \textit{R. Dietmann} [Arch. Math. 75, 195-197 (2000; Zbl 1034.11004)], it is shown that one may take \(\theta= (s+1)/2\) if \(F\) is a diagonal cubic form, and \(s\) is odd. One approach to this question uses linear subspaces on which \(F\) vanishes modulo \(m\), and various results on the dimension of such spaces are obtained. NEWLINENEWLINENEWLINEThis final result gives bounds for NEWLINE\[NEWLINE\min_{0<|x|\leq N}\|\lambda_1x_1^k+\cdots+ \lambda_s x_s^k\|,NEWLINE\]NEWLINE improving the previous paper [Mathematika 30, 164-188 (1983; Zbl 0532.10011)] in this series.