On Artin's Conjecture: Pairs of Additive Forms

Abstract

It is established that for every pair of additive forms $f={\sum }_{i=1}^s{a}_i{x}_i^k,g={\sum }_{i=1}^s{b}_i{x}_i^k$ of degree $k$ in $s>2k^2$ variables the equations $f=g=0$ have a non-trivial $p$-adic solution for all odd primes $p$.