Low, Lewis; Pitman, Jane; Wolff, Alison Simultaneous diagonal congruences. (English) Zbl 0643.10011 J. Number Theory 29, No. 1, 31-59 (1988). 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 Cited in 2 ReviewsCited in 8 Documents 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 Keywords:simultaneous congruences; simultaneous equations; partitionable matrices; diagonal integral forms; lower bound; number of solutions; p-adic solution Citations:Zbl 0207.353; Zbl 0541.10024 PDFBibTeX XMLCite \textit{L. Low} et al., J. Number Theory 29, No. 1, 31--59 (1988; Zbl 0643.10011) Full Text: DOI References: [1] Aigner, M., (Combinatorial Theory (1979), Springer-Verlag: Springer-Verlag New York/Heidelberg/Berlin) · Zbl 0415.05001 [2] Borevich, Z. I.; Shafarevich, I. R., (Number Theory (1966), Academic Press: Academic Press New York) · Zbl 0145.04902 [3] Chowla, S.; Shimura, G., On the representation of zero by a linear combination of \(k\) th powers, Norske Vid. Selsk. Forh. (Trondheim), 36, 169-176 (1963) · Zbl 0119.04405 [4] Cook, R. J., Pairs of additive equations, Michigan Math. J., 19, 325-331 (1972) · Zbl 0244.10046 [5] Davenport, H.; Lewis, D. J., Simultaneous equations of additive type, Philos. Trans. Roy. Soc. London Ser. A, 264, 557-595 (1969) · Zbl 0207.35304 [6] Edmonds, J., Minimum partition of a matroid into independent subsets, J. Res. Nat. Bur. Standards, 69B, 67-72 (1965) · Zbl 0192.09101 [7] Lloyd, D. P., Bounds for Solutions of Diophantine Equations, (Ph.D. thesis (1975), University of Adelaide: University of Adelaide Adelaide, South Australia) [8] Nadesalingam, T., Diophantine Inequalities in Many Variables, (Ph.D. thesis (1981), University of Adelaide: University of Adelaide Adelaide, South Australia) · Zbl 0654.10021 [9] Pitman, J., Bounds for solutions of diagonal inequalities, Acta Arith., 18, 179-190 (1971) · Zbl 0225.10024 [10] Schmidt, W. M., Simultaneous \(p\)-adic zeros of quadratic forms, Mh. Math., 90, 45-65 (1980) · Zbl 0431.10013 [11] Schmidt, W. M., The solubility of certain \(p\)-adic equations, J. Number Theory, 19, 63-80 (1984) · Zbl 0541.10024 [12] Tietäväinen, A., On a problem of Chowla and Shimura, J. Number Theory, 3, 247-252 (1971) · Zbl 0213.05203 [13] Toliver, R. H., Bounds for Solutions of Two Simultaneous Additive Equations of Odd Degree, (Ph.D. thesis (1975), University of Michigan) · Zbl 0677.10037 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.