#### Sabancı University Mathematics Colloquium

Permutations of the form $x ^ { k } - \gamma \operatorname { Tr } ( x )$ and curves over finite fields
Nurdagül Anbar
Sabancı University, Turkey
Özet : Let $q$ be a power of a prime $p$, and let $\mathbb { F } _ { q }$ be the finite field with $q$ elements. A polynomial $P ( x ) \in \mathbb { F } _ { q } [ x ]$ is called a permutation of $\mathbb { F } _ { q }$ if the associated map from $\mathbb { F } _ { q }$ to $\mathbb { F } _ { q }$ defined by $x \mapsto P ( x )$ is a bijection, i.e., it permutes the elements of $\mathbb { F } _ { q }$. In this talk, we consider the polynomials of the form $P ( x ) = x ^ { k } - \gamma \operatorname { Tr } ( x )$ over $\mathbb { F } _ { q ^n}$ for $n \geq 2$, where $\mathbb { F } _ { q ^n}$ is the extension of $\mathbb { F } _ { q }$ of degree $n$ and Tr is the absolute trace from $\mathbb { F } _ { q ^n}$ to $\mathbb { F } _ { q }$. We show that $P(x)$ is not a permutation of $\mathbb { F } _ { q ^n}$ in the case gcd$(k, q^n − 1) > 1$. Our proof uses an absolutely irreducible curve over $\mathbb { F } _ { q ^n}$ and the number of rational points on it.
 Tarih : 04.04.2019 Saat : 13:40 Yer : FENS G035 Dil : English