Pairs of diagonal forms of degree 3τ.2 and Artin's conjecture

Abstract

One version of Artin's Conjecture states that for a pair of diagonal forms of degree k, with integer coefficients, there exist nontrivial common p-adic zeros provided the number of variables is greater than 2k2. This version of the conjecture is known to be true for every degree k with the possible exception of degrees of the form pτ(p−1), when p is a prime number. In this paper, we show that the conjecture is true for k=3τ⋅2, giving an indication that the conjecture may be true even for these critical degrees.