## View abstract

### Session S26 - Finite fields and applications

Thursday, July 15, 12:30 ~ 12:50 UTC-3

## Rational functions with Small Value Set

### Luciane Quoos

Let $q$ be a power of a prime $p$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For any rational function $h(x) \in \mathbb{F}_q(x)$, its value set is defined as $$V_h =\{ h(\alpha) \mid \alpha \in \mathbb{P}^1(\mathbb{F}_q) \}\subset \mathbb{P}^1(\mathbb{F}_q)=\mathbb{F}_q \cup \{ \infty \} .$$ If $h(x) \in \mathbb{F}_q(x)$ is a rational function of degree $d$, then one has the trivial bound $$\left \lceil \frac{q+1}{d} \right \rceil \leqslant \# V_h \leqslant q+1.$$
In connection with Galois Theory and Algebraic Curves, under certain hypothesis, it is proved that a function $h(x) \in \mathbb{F}_q(x)$ having a "small value set" is equivalent to the field extension $\mathbb{F}_q(x)/\mathbb{F}_q(h(x))$ being Galois.