On Artin's conjecture for pairs of diagonal forms

TL;DR

This paper investigates solutions to pairs of diagonal forms over p-adic fields, establishing bounds on the number of variables needed for the existence of non-trivial solutions across various primes and exponents.

Contribution

It provides new bounds on the number of variables ensuring p-adic solutions for pairs of diagonal forms, extending Artin's conjecture to broader cases.

Findings

Non-trivial p-adic solutions exist for s above certain bounds.

Bounds depend on prime p, exponent τ, and form degree d.

Results cover various ranges of τ and p with explicit constants.

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 $\tau = 1, p \ge 5$.