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#### Full classification of permutation rational functions and complete rational functions of degree three over finite fields

##### External Resource

https://doi.org/10.1007/s10623-020-00715-0

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##### Fulltext (public)

arXiv:1805.03097.pdf

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##### Citation

Ferraguti, A., & Micheli, G. (2020). Full classification of permutation rational
functions and complete rational functions of degree three over finite fields.* Designs, Codes and Cryptography,*
*88*(5), 867-886. doi:10.1007/s10623-020-00715-0.

Cite as: https://hdl.handle.net/21.11116/0000-0006-9CD8-7

##### Abstract

Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$ and

$\mathbb F_q(x)$ be the field of rational functions over $\mathbb F_q$. In this

paper we classify all rational functions $\varphi\in \mathbb F_q(x)$ of degree

3 that induce a permutation of $\mathbb P^1(\mathbb F_q)$. Our methods are

constructive and the classification is explicit: we provide equations for the

coefficients of the rational functions using Galois theoretical methods and

Chebotarev Density Theorem for global function fields. As a corollary, we

obtain that a permutation rational function of degree 3 permutes $\mathbb F_q$

if and only if it permutes infinitely many of its extension fields. As another

corollary, we derive the well-known classification of permutation polynomials

of degree 3. As a consequence of our classification, we can also show that

there is no complete permutation rational function of degree $3$ unless $3\mid

q$ and $\varphi$ is a polynomial.

$\mathbb F_q(x)$ be the field of rational functions over $\mathbb F_q$. In this

paper we classify all rational functions $\varphi\in \mathbb F_q(x)$ of degree

3 that induce a permutation of $\mathbb P^1(\mathbb F_q)$. Our methods are

constructive and the classification is explicit: we provide equations for the

coefficients of the rational functions using Galois theoretical methods and

Chebotarev Density Theorem for global function fields. As a corollary, we

obtain that a permutation rational function of degree 3 permutes $\mathbb F_q$

if and only if it permutes infinitely many of its extension fields. As another

corollary, we derive the well-known classification of permutation polynomials

of degree 3. As a consequence of our classification, we can also show that

there is no complete permutation rational function of degree $3$ unless $3\mid

q$ and $\varphi$ is a polynomial.