CHARACTERIZING CONGRUENCE PRESERVING FUNCTIONS Z/nZ → Z/mZ VIA RATIONAL POLYNOMIALS

Abstract : We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $\Z/n\Z\to\Z/m\Z$. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the functions $\lcm(k)\,P_k$ where $\lcm(k)$ is the least common multiple of $2,\ldots,k$ (viewed in $\Z/m\Z$). As a consequence, when $n\geq m$, the number of such functions is independent of $n$.
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Submitted on : Monday, January 25, 2016 - 6:08:55 PM
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Serge Grigorieff, Irene Guessarian, Patrick Cégielski. CHARACTERIZING CONGRUENCE PRESERVING FUNCTIONS Z/nZ → Z/mZ VIA RATIONAL POLYNOMIALS. 2016. ⟨hal-01260934⟩

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