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.

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).

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