Simultaneous diagonal congruences
Considering R diagonal integral forms of degree k in n variables, \[ F_ i({\mathfrak x})=F_ i=a_{i1}x^ k_ 1+...+a_{n1}x^ k_ n\quad (i=1,...,R), \] \textit{H. Davenport} and \textit{D. J. Lewis} [Philos. Trans. R. Soc. Lond., Ser. A 264, 557-595 (1969; Zbl 0207.353)] showed that the conditions \(n\geq [9R^ 2k \log 3Rk]\) for odd k, and \(n\geq [48R^ 2k^ 3 \log 3Rk^ 2]\) for even k, ensure the existence of a primitive solution of the congruences \[ (1)\quad F_ 1\equiv...\equiv F_ R\equiv 0 (mod p^ s) \] for all prime powers \(p^ s\) if the forms are suitably normalized. Furthermore, they showed that these conditions on n also guarantee the existence of a non-trivial p-adic solution of the equations \((2)\quad F_ 1=...=F_ R=0.\) In the present paper the authors first present conditions for partitioning the coefficient matrix \([a_{ij}]\) and then obtain explicitly a lower bound for the number of solutions of (1) with odd k. As a consequence they prove that if \(n\geq 2R^ 2k \log k\) for large odd k or \(n\geq [48Rk^ 3 \log 3Rk^ 2]\) for any \(k\geq 3\), then there exists a non-trivial p-adic solution of (2) for each prime p. An interesting discussion on the relationship of the authors' methods to the work of Davenport and Lewis [ibid.] and \textit{W. M. Schmidt} [J. Number Theory 19, 63-80 (1984; Zbl 0541.10024)] is given in the paper.
scientific article; zbMATH DE number 4047801
scientific article