Variations on the Arkhipov-Karačuba Type Counterexamples to Artin's Conjecture

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 $d\ge 4$ were later discovered. The major types of counterexamples are Terjanian Type and Arkhipov-Karačuba 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-Karačuba 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.

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