Characterizing congruence preserving functions Z/nZ→ Z/mZ via rational polynomials

2015/05/30 by Patrick Cegielski, Patrick Cégielski, Serge Grigorieff +5
Computer Science · Mathematics · #Advanced Algebra and Logic #Commutative Algebra and Its Applications #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #cs.DM #math.NT

paper · pdf · doi:10.48550/arxiv.1506.00133

arxiv created 2015/05/30 · openalex publication_date 2015/05/30 · arxiv updated 2015/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a basis of rational polynomial-like functions P₀,…,P_n-1 for the free module of functions Z/nZ→ 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,…,k (viewed in Z/mZ). As a consequence, when n≥ m, the number of such functions is independent of n.

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