## 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

#### Universidade Federal do Rio de Janeiro, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak837ccdfbfd5a979a6e74dff69e6b8f53').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy837ccdfbfd5a979a6e74dff69e6b8f53 = 'l&#117;c&#105;&#97;n&#101;' + '&#64;'; addy837ccdfbfd5a979a6e74dff69e6b8f53 = addy837ccdfbfd5a979a6e74dff69e6b8f53 + '&#105;m' + '&#46;' + '&#117;frj' + '&#46;' + 'br'; var addy_text837ccdfbfd5a979a6e74dff69e6b8f53 = 'l&#117;c&#105;&#97;n&#101;' + '&#64;' + '&#105;m' + '&#46;' + '&#117;frj' + '&#46;' + 'br';document.getElementById('cloak837ccdfbfd5a979a6e74dff69e6b8f53').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy837ccdfbfd5a979a6e74dff69e6b8f53 + '\'>'+addy_text837ccdfbfd5a979a6e74dff69e6b8f53+'<\/a>';

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.

Joint work with Daniele Bartoli (Università degli Studi di Perugia, Italy) and Herivelto Borges (Universidade de São Paulo, Brazil).