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

## TL;DR

This paper provides a more direct universal first-order definition of the polynomial ring _q[t] within its field of fractions _q(t), using a minimal number of quantifiers in a logical language.

## Contribution

It introduces a new, more straightforward universal definition of _q[t] in _q(t) with fewer quantifiers, improving on existing literature.

## Key findings

- Universal definition uses 89 quantifiers, more direct than previous.
- Modified to a parameter-free universal definition with 90 quantifiers.
- Assumes odd characteristic of _q for the definitions.

## 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.