On Artin’s conjecture for pairs of diagonal forms

Abstract

Let [Formula: see text] be an odd prime and [Formula: see text]. In the spirit of Artin’s conjecture, consider the system of two diagonal forms of degree [Formula: see text] in [Formula: see text] variables given by [Formula: see text] [Formula: see text] with [Formula: see text]. For [Formula: see text], this paper shows that this system has a non-trivial [Formula: see text]-adic solution for every [Formula: see text], and for every [Formula: see text], where [Formula: see text]. Moreover, for [Formula: see text], this system will have a non-trivial [Formula: see text]-adic solution for every [Formula: see text].