On p-Adic Zeros of Additive Forms of Even Degree

Hemar Godinho

In this paper it is proved that the pair f = a_{1}x_{1}^{k} + ⋯ + a_{n}x_{n}^{k}, g = b_{1}x_{1}^{k} + ⋯ + b_{n}x_{n}^{k}, with k = p^{τ} (p − 1)k_{0} and (k_{0}, p) = 1, has a common p-adic zero provided n ≥ 2k^{2+w(p,τ)}, where w(p, τ) = 1/(log p + \frac{1}{τ} log(p − 1)). It is also proved that if n ≥ 2k^{2}, then the pair f, g has a common p-adic zero provided that k_{0} ≥ 2^{τ}.