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.