Artin's Conjecture and systems of diagonal equations

TL;DR

This paper demonstrates that Artin's conjecture, which predicts p-adic solvability of Diophantine equations, does not hold for infinitely many systems of multiple homogeneous diagonal equations.

Contribution

It provides a counterexample to Artin's conjecture for systems of diagonal equations when the number of equations exceeds one.

Findings

Artin's conjecture fails for infinitely many systems

Counterexamples exist for r>1

p-adic solubility does not always hold

Abstract

We show that Artin's conjecture concerning p-adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r>1.