Pairs of additive forms of degree $p^\tau(p-1)$

Abstract

Let \[ f(x_1,...,x_n)=a_{1}x_{1}^{k}+\cdots+a_{n}x_{n}^{k}\\ g(x_1,...,x_n)=b_{1}x_{1}^{k}+\cdots+b_{n}x_{n}^{k} \] be a pair of additive forms of degree $k=p^{\tau}(p-1)$. We are interested in finding conditions which guarantee the existence of $p$-adic zeros for this pair of forms. A well-known conjecture due to Emil Artin states that the condition $n > 2k^2$ is sufficient. Here we prove that $$n > 2\left(\frac{p}{p-1}\right)k^2-2k$$ is sufficient, provided that $p > 5$ and $\tau \geq \dfrac{p-1}{2}$.