Finite polynomial orbits in finitely generated domains.(English)Zbl 0897.13024

Let $$R$$ be a commutative domain of characteristic $$0$$. A polynomial orbit in $$R$$ is a set $$O_f(a)=\{ a_j: j=0,1,2,\ldots\}$$, where $$a_0\in R$$ and $$a_{j+1}=f(a_j)$$ for $$j=0,1,2,\ldots$$, for some non-zero polynomial $$f\in R[X]$$. Such an orbit is called finite of length $$k$$ if there are indices $$j<k$$ with $$a_j=a_k$$ and if $$k$$ with this property is minimal. Such an orbit is called a polynomial cycle of length $$k$$ if there is an index $$k>0$$ with $$a_k=a_0$$ and if $$k$$ is minimal. Northcott showed that if $$R$$ is the ring of integers of a number field, then for given $$f$$, if an orbit $$O_f(a)$$ has finite length, then this length is bounded above by a function of $$R$$ and $$f$$, and Narkiewicz showed that the dependence on $$f$$ is unnecessary. The authors prove a more general and more precise result.
Suppose that the domain $$R$$ has the following two properties: (i) every polynomial cycle in $$R$$ has length $$\leq B(R)$$; (ii) the equation $$x_1+x_2+x_3=1$$ has only finitely many solutions with $$x_i\in R^*$$ and $$x_i\not= 1$$ for $$i=1,2,3$$, where $$R^*$$ denotes the group of invertible elements of $$R$$. Denote this number of solutions by $$C(R)$$. It is known that such quantities $$B(R)$$ and $$C(R)$$ exist for every domain $$R$$ that is finitely generated over $${\mathbb{Z}}$$ and for several rings $$R$$, explicit upper bounds for $$B(R)$$ have been computed [cf. F. Halter-Koch and W. Narkiewicz, Monatsh. Math. 119, No. 4, 275-279 (1995; Zbl 0840.13003) for $$B(R)$$ and H. P. Schlickewei, Invent. Math. 102, No. 1, 95-107 (1990; Zbl 0711.11017) and J.-H. Evertse, Invent. Math. 122, No. 3, 559-601 (1995; Zbl 0851.11019) for $$C(R)$$].
The authors show the following general result: $$D(R)\leq {1\over 3}(31+C(R))-1$$. Using the explicit bounds mentioned above, the authors show among others that $$D({\mathbb{Z}})=4$$ and that $$D(R_K)\leq {2\over 3}d4^d(31+2^{1031d})$$ if $$R_K$$ is the ring of integers of a number field $$K$$ of degree $$d$$. They give a more general result for rings of $$S$$-integers.

MSC:

 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13B25 Polynomials over commutative rings 11S05 Polynomials 11R09 Polynomials (irreducibility, etc.) 13E15 Commutative rings and modules of finite generation or presentation; number of generators 13G05 Integral domains
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References:

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