New Results on Additive Forms of Even Degree

Hemar Godinho

This paper presents many recent results on pairs of additive forms of even degree, outlining the main techniques involved, and the proof of the following result: A pair of additive forms of degree k, with k = p^{τ} (p − 1)k_{0} and (k_{0}, p) = 1, has a common p-adic zero provided the number of variables n ≥ 2k^{2+w(p,τ)}, where w(p, τ) = 1/(log p + \frac{1}{τ} log(p − 1)).