Conditions for the Solvability of Systems of Two and Three Additive Forms Over p ‐Adic Fields

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 ax k + by k + cz k ≡ d (mod p ) has a solution with xyz ≢ 0(mod p ) we have proved that any system of two additive forms of odd degree k with at least 6 k + 1 variables, and any system of three additive forms of odd degree k with at least 14 k + 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 > k 4 . 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.