On Artin's conjecture for pairs of diagonal forms

Abstract

Let $p$ be an odd prime and $d=p^{\tau }(p-1)$. In the spirit of Aritn's conjecture, consider the system of two diagonal forms of degree $d$ in $s$ variables given by \begin{equation*}\begin{split} a_1x_1^d + \cdots + a_sx_s^d = 0\\ b_1x_1^d + \cdots + b_sx_s^d = 0 \end{split} \end{equation*} with $a_i,b_i\in {\mathbb{Q}}_p$. For $s>2\frac{p}{p-1}{d}^2-2d$, this paper shows that this system has a non-trivial $p$-adic solution for every $\tau \ge 3,p\ge 7$, and for every $\tau =2,p\ge \frac{C}{2}+4$, where $C\le 9996$. Moreover, for $s>(2\frac{p}{p-1}+\frac{C-3}{2p-2})d^2-2d$, this system will have a non-trivial $p$-adic solution for every $\tau =1,p\ge 5$.

References in corpus (1)

• On Artin's Conjecture: Pairs of Additive Forms