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Hemar Godhino; Paulo H. A. Rodrigues
On ${p}$-adic zeros of systems of diagonal forms restricted by a congruence condition
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 205-219, doi: 10.5802/jtnb.582
Article PDF | Reviews MR 2332062 | Zbl 1131.11023

Résumé - Abstract

This paper is concerned with non-trivial solvability in $p$-adic integers of systems of additive forms. Assuming that the congruence equation $ax^k+by^k+cz^k \equiv d \,(\mbox {mod}\,p)$ has a solution with $xyz \nequiv0 \,(\mbox {mod}\,p)$ we have proved that any system of $R$ additive forms of degree $k$ with at least $2\cdot 3^{R-1}\cdot k +1$ variables, has always non-trivial $p$-adic solutions, provided $p \nmid k$. The assumption of the solubility of the above congruence equation is guaranteed, for example, if $p > k^4$.

Bibliography

[1] O.D. Atkinson, J. Brüdern, R.J. Cook, Simultaneous additive congruences to a large prime modulus. Mathematika 39 (1) (1992), 1–9.  MR 1176464 |  Zbl 0774.11016
[2] H. Davenport, D.J. Lewis, Simultaneous equations of additive type. Philos. Trans. Roy. Soc. London, Ser. A 264 (1969), 557–595.  MR 245542 |  Zbl 0207.35304
[3] H. Godinho, P. H. A. Rodrigues, Conditions for the solvability of systems of two and three additive forms over p-adic fields. Proc. of the London Math. Soc. 91 (2005), 545–572.  MR 2180455 |  Zbl 1086.11020
[4] D.J. Lewis, H. Montgomery, On zeros of p-adic forms. Michigan Math. Journal 30 (1983), 83–87. Article |  MR 694931 |  Zbl 0531.10026
[5] L. Low, J. Pitman, A. Wolff, Simultaneous Diagonal Congruences. J. Number Theory 29 (1988), 31–59.  MR 938869 |  Zbl 0643.10011
[6] I. D. Meir, Pairs of Additive Congruences to a Large Prime Modulus. J. Number Theory 63 (1997), 132–142.  MR 1438653 |  Zbl 0871.11024