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Classification of polynomial mappings between commutative groups. (English) Zbl 1311.20052

If \(f(X)\) is an integral-valued polynomial with rational coefficients and \(m,n\) are positive integers, then \(\Phi\colon a\mapsto f(a)\bmod n\) is a map from \(\mathbb Z\) to the cyclic group \(\mathbb Z_n\). If it happens that the value \(\Phi(a)\) depends only on the residue \(a\bmod m\), then \(\Phi\) induces a map \(\Psi\colon\mathbb Z_m\to\mathbb Z_n\) of cyclic groups. If \(A,B\) are arbitrary finite Abelian groups, then writing them as direct sums of the same number \(r\) of cyclic groups (with some trivial factors, if necessary) one can obtain in a similar way mappings \(A\to B\) induced by integral-valued polynomials in \(r\) variables. The author studies properties of the mappings obtained in this way and presents their classification.

MSC:

20K01 Finite abelian groups
12E99 General field theory
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11T06 Polynomials over finite fields
08A40 Operations and polynomials in algebraic structures, primal algebras
41A05 Interpolation in approximation theory
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References:

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