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- W4300703605 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 $varphiin mathbb F_q(x)$ 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 $3mid q$ and $varphi$ is a polynomial." @default.
- W4300703605 created "2022-10-04" @default.
- W4300703605 creator A5050777736 @default.
- W4300703605 creator A5071321653 @default.
- W4300703605 date "2018-05-08" @default.
- W4300703605 modified "2023-09-26" @default.
- W4300703605 title "Full Classification of permutation rational functions and complete rational functions of degree three over finite fields" @default.
- W4300703605 doi "https://doi.org/10.48550/arxiv.1805.03097" @default.
- W4300703605 hasPublicationYear "2018" @default.
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