On permutation binomials of index $q^{e-1}+q^{e-2}+\cdots+1$

Abstract: The permutation binomial $f(x) = x^r(x^{q-1} + A)$ was studied by K. Li, L. Qu, and X. Chen over $\mathbb{F}_{q^2}$. They found that for $1 \leq r \leq q+1$, $f(x)$ is a permutation binomial if and only if $r = 1$. Over the finite field $\mathbb{F}_{q^3}$ of odd characteristic, X. Liu obtained an analogous result, in which for $1 \leq r \leq q^2+q+1$, $f(x)$ permutes $\mathbb{F}_{q^3}$ if and only if $r = 1$. In this investigation, we complete the characterization for $f(x)$ over both $\mathbb{F}_{q^2}$ and $\mathbb{F}_{q^3}$, as well as obtain a complete characterization over $\mathbb{F}_{q^4}$. Furthermore, for $e \geq 5$, we present some partial results which narrow down considerably the search for $r's$ that do indeed yield permutation binomials of the form $f(x) = x^r(x^{q-1} + A)$ over $\mathbb{F}_{q^e}$.

Joint work with Ariane Masuda and Ivelisse Rubio.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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