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- W4387356175 abstract "We study the class of univariate polynomials $beta_k(X)$, introduced by Carlitz, with coefficients in the algebraic function field $mathbb F_q(t)$ over the finite field $mathbb F_q$ with $q$ elements. It is implicit in the work of Carlitz that these polynomials form a $mathbb F_q[t]$-module basis of the ring $text{Int}(mathbb F_q[t]) = {f in mathbb F_q(t)[X] mid f(mathbb F_q[t]) subseteq mathbb F_q[t]}$ of integer-valued polynomials on the polynomial ring $mathbb F_q[t]$. This stands in close analogy to the famous fact that a $mathbb Z$-module basis of the ring $text{Int}(mathbb Z)$ is given by the binomial polynomials $binom{X}{k}$. We prove, for $k = q^s$, where $s$ is a non-negative integer, that $beta_k$ is irreducible in $text{Int}(mathbb F_q[t])$ and that it is even absolutely irreducible, that is, all of its powers $beta_k^m$ with $m>0$ factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that $beta_k$ is not even irreducible if $k$ is not a power of $q$." @default.
- W4387356175 created "2023-10-05" @default.
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- W4387356175 date "2023-10-03" @default.
- W4387356175 modified "2023-10-06" @default.
- W4387356175 title "Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields" @default.
- W4387356175 doi "https://doi.org/10.48550/arxiv.2310.02061" @default.
- W4387356175 hasPublicationYear "2023" @default.
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