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


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
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