Variations on the Arkhipov-Kara\v{c}uba Type Counterexamples to Artin's Conjecture

TL;DR

This paper constructs new counterexamples to Artin's conjecture over p-adic fields, specifically odd-degree counterexamples for primes greater than 3, and discusses methods to increase the number of variables.

Contribution

It introduces modifications to Arkhipov-Karačuba counterexamples to produce odd-degree counterexamples and explores increasing variable counts.

Findings

Constructed counterexamples with odd degrees divisible by (p-1)/2 for primes > 3.

Extended known counterexamples to include primes congruent to 3 mod 4.

Proposed ideas for increasing the number of variables in counterexamples.

Abstract

It was conjectured by Emil Artin in the 1930's that every $d$-form $F(x_1, x_2, $\ldots$, x_n)$ over the $p$-adic field in more than $d^2$ variables has a solution that is not $(0, 0, \cdots, 0)$ (non-trivial solution) over the $p$-adic field. This is true for $d=2$ and $d=3$. However, many counterexamples for $d \geq 4$ were later discovered. The major types of counterexamples are Terjanian Type and Arkhipov-Kara\v{c}uba Type. The degrees of all known counterexamples, however, are divisible by $p-1$, which means that they are even for all odd primes. In this article we apply modifications to the known Arkhipov-Kara\v{c}uba Type counterexamples to construct counterexamples with odd degrees that are divisible by $\frac{p-1}2$ for all primes greater than $3$ and congruent to $3$ modulo $4$ and then propose some ideas about increasing the number of variables in the counterexamples.