Rational periodic points of rational functions.

*(English)*Zbl 0819.11045Let \(\varphi\) be a rational map of degree \(\geq 2\) of the projective line over a finite extension of a \(p\)-adic field \(K\) into itself and assume that one can write it in the form \(\varphi= [\varphi_ 1, \varphi_ 2]\) where \(\varphi_ i\) are homogeneous polynomials whose resultant is a unit of \(K\) (i.e. \(\varphi\) has good reduction). Let \(p^ f\) be the cardinality of the corresponding residue class field.

The authors show that if \(\varphi\) has a periodic point of minimal period \(n\), then \(n\) has the form \(abp^ c\), where \(a\leq p^ f+ 1\), \(b\) divides \(p^ f-1\) and \(c\geq 0\). This enables them to show that minimal periods of periodic points for similar rational maps defined over a number field \(L\) of degree \(N\) over the rationals do not exceed \((12(t+2) \log(5t+ 10))^{4N}\), with \(t\) denoting the number of primes of \(L\) at which \(\varphi\) has bad reduction, improving thus essentially upon a previous bound [the reviewer, Colloq. Math. 58, 151-155 (1989; Zbl 0703.12002)]. As an example \(Q\)-rational periodic points of \((ax^ 2+ bx+ c)/ x^ 2)\) are studied and it is shown that in this case there are no periodic points with minimal periods exceeding 3.

In an addendum the authors point out that in case of polynomial maps their bounds have been improved by T. Pezda [Acta Arith. 66, 11-22 (1994; Zbl 0803.11063)].

Reviewer’s remark: In the introduction the authors attribute to the reviewer an assertion (boundedness of the number of periodic points of a polynomial map \(\varphi\) defined over \(\mathbb{Q}\) depending only on the number of prime divisors of the set of denominators of coefficients of \(\varphi\)), which I never stated and which in fact is trivially false, as the example \((x-1) (x-2) \cdots (x-n)+ x\) shows.

The authors show that if \(\varphi\) has a periodic point of minimal period \(n\), then \(n\) has the form \(abp^ c\), where \(a\leq p^ f+ 1\), \(b\) divides \(p^ f-1\) and \(c\geq 0\). This enables them to show that minimal periods of periodic points for similar rational maps defined over a number field \(L\) of degree \(N\) over the rationals do not exceed \((12(t+2) \log(5t+ 10))^{4N}\), with \(t\) denoting the number of primes of \(L\) at which \(\varphi\) has bad reduction, improving thus essentially upon a previous bound [the reviewer, Colloq. Math. 58, 151-155 (1989; Zbl 0703.12002)]. As an example \(Q\)-rational periodic points of \((ax^ 2+ bx+ c)/ x^ 2)\) are studied and it is shown that in this case there are no periodic points with minimal periods exceeding 3.

In an addendum the authors point out that in case of polynomial maps their bounds have been improved by T. Pezda [Acta Arith. 66, 11-22 (1994; Zbl 0803.11063)].

Reviewer’s remark: In the introduction the authors attribute to the reviewer an assertion (boundedness of the number of periodic points of a polynomial map \(\varphi\) defined over \(\mathbb{Q}\) depending only on the number of prime divisors of the set of denominators of coefficients of \(\varphi\)), which I never stated and which in fact is trivially false, as the example \((x-1) (x-2) \cdots (x-n)+ x\) shows.

Reviewer: W.Narkiewicz (Wrocław)