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Simultaneous diagonal congruences. (English) Zbl 0643.10011

Considering R diagonal integral forms of degree k in n variables, \[ F_ i({\mathfrak x})=F_ i=a_{i1}x^ k_ 1+...+a_{n1}x^ k_ n\quad (i=1,...,R), \] H. Davenport and 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 W. M. Schmidt [J. Number Theory 19, 63-80 (1984; Zbl 0541.10024)] is given in the paper.
Reviewer: M.-C.Liu

MSC:

11D72 Diophantine equations in many variables
11P05 Waring’s problem and variants
11D41 Higher degree equations; Fermat’s equation
11D88 \(p\)-adic and power series fields
11E95 \(p\)-adic theory
11E76 Forms of degree higher than two
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References:

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