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H. Godinho, P. H. A. Rodrigues, Conditions for the Solvability of Systems of Two and Three Additive Forms Over p-Adic Fields, Proceedings of the London Mathematical Society, Volume 91, Issue 3, November 2005, Pages 545–572, https://doi.org/10.1112/S0024611505015455
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Abstract
This paper is concerned with non-trivial solvability in p-adic integers of systems of two and three additive forms. Assuming that the congruence equation axk + byk + czk ≡ d (modp) has a solution with xyz ≢ 0(modp) we have proved that any system of two additive forms of odd degree k with at least 6k + 1 variables, and any system of three additive forms of odd degree k with at least 14k + 1 variables always has non-trivial p-adic solutions, provided p does not divide k. The assumption of the solubility of the congruence equation above is guaranteed for example if p > k4.
In the particular case of degree k = 5 we have proved the following results. Any system of two additive forms with at least n variables always has non-trivial p-adic solutions provided n ≥ 31 and p > 101 or n ≥ 36 and p > 11. Furthermore any system of three additive forms with at least n variables always has non-trivial p-adic solutions provided n ≥ 61 and p > 101 or n ≥ 71 and p > 11. 2000 Mathematics Subject Classification 11D72, 11D79.
