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Polynomial cycles in algebraic number fields. (English) Zbl 0703.12002
Let $$K$$ be a field, $$P\in K[x]$$. A sequence $$x_0,x_1,\ldots,x_{n-1}$$ of distinct elements of $$K$$ is an $$n$$-cycle of $$P$$ provided $$P(x_{n-1})=x_0$$ and for $$i=0,1,\ldots,(n-2)$$ one has $$P(x_i)=x_{i+1}$$. The author proves that if $$P$$ is a monic polynomial, whose coefficients are integers of $$K$$, then the length $$n$$ of its cycles lying in $$K$$ is bounded by a constant, which depends only on the degree $$M$$ of $$K$$, but not on $$P$$ nor on $$K$$ itself. This constant does not exceed $$\exp(C2^M)$$ with a suitable absolute constant $$C$$. (In fact the author proves a more elaborate theorem of which the result quoted above is a corollary.)
Reviewer: I. N. Baker

MSC:
 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
Keywords:
cyclic points; monic polynomial
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