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arXiv · 1506.00133

Characterizing congruence preserving functions $Z/nZ\to Z/mZ$ via rational polynomials

Patrick Cegielski·Serge Grigorieff·Irene Guessarian
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Abstract

We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $Z/nZ\to Z/mZ$. 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 $Z/mZ$). As a consequence, when $n\geq m$, the number of such functions is independent of $n$.

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Patrick Cegielski, Serge Grigorieff, Irene Guessarian. 2015-05-30. Characterizing congruence preserving functions $Z/nZ\to Z/mZ$ via rational polynomials. https://arxiv.org/abs/1506.00133

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