On Artin's Conjecture: Pairs of Additive Forms

TL;DR

This paper proves that for pairs of additive forms with sufficiently many variables, the equations have non-trivial p-adic solutions for all odd primes, advancing understanding of Artin's conjecture in additive number theory.

Contribution

It establishes the existence of non-trivial p-adic solutions for pairs of additive forms with more than 2k^2 variables, confirming a case of Artin's conjecture.

Findings

Non-trivial p-adic solutions exist for all odd primes p.

The result applies to pairs of additive forms with s > 2k^2 variables.

The paper extends previous results on additive forms and p-adic solvability.

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 $p$.