### Cycles of polynomial mappings in several variables.

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We find all possible cycle-lengths for polynomial mappings in two variables over rings of integers in quadratic extensions of rationals.

1. Let R be a domain and f ∈ R[X] a polynomial. A k-tuple $x\u2080,x\u2081,...,{x}_{k-1}$ of distinct elements of R is called a cycle of f if $f\left({x}_{i}\right)={x}_{i+1}$ for i=0,1,...,k-2 and $f\left({x}_{k-1}\right)=x\u2080$. The number k is called the length of the cycle. A tuple is a cycle in R if it is a cycle for some f ∈ R[X]. It has been shown in [1] that if R is the ring of all algebraic integers in a finite extension K of the rationals, then the possible lengths of cycles of R-polynomials are bounded by the number ${7}^{7\xb7{2}^{N}}$, depending only on the degree N of K. In this note we consider...

It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer ${N}_{r}$ such that the polynomial $f\left(X\right)/{N}_{r}$ represents at least r distinct primes.

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