Polynomial orbits in finite commutative rings

Polynomial functions on finite commutative rings

Every function on a nite residue class ring D=I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a nite commutative local ring with maximal ideal P of nilpotency N satisfying for all a; b 2 R, if ab 2 Pn then a 2 P k , b 2 P j with k + j min(n;N), we determine the number of functions (as...

Finite Commutative Rings

‡ Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b∈R, if ab∈ P then a∈ P k , b∈ P j with k+ j ≥min(n,N), we determine the number of functions (as...

Commutative Rings and Finite Fields

Polynomial Equivalence of Finite Rings

We prove that Zpn and Zp[t]/(t) are polynomially equivalent if and only if n ≤ 2 or p = 8. For the proof, employing Bernoulli numbers, we provide the polynomials which compute the carry-on part for the addition and multiplication in base p. As a corollary, we characterize finite rings of p elements up to polynomial equivalence.

Diophantine properties of finite commutative rings

Simple observations on diophantine definability over finite commutative rings lead to a characterization of those rings in terms of their diophantine behavior. A.M.S. Classification: 13M10, 11T06, 03G99.