A New Universal Definition of $\mathbb{F}_q [t]$ in $\mathbb{F}_q (t)$

Abstract

This paper gives a universal definition of $\mathbb{F}_q [t]$ in $\mathbb{F}_q (t)$ using 89 quantifiers, more direct than those that exist in the current literature. The language $\mathcal{L}_{\mbox{rings}, t}$ we consider here is the language of rings $\{0, 1, +, -, \cdot\}$ with an additional constant symbol $t$. We then modify this definition marginally to universally define $\mathbb{F}_q [t]$ in $\mathbb{F}_q (t)$ without parameters, using 90 quantifiers. We assume throughout that the characteristic of $\mathbb{F}_q$ is odd.