During work on the article “Scarcity of cycles for rational functions over a number field”, Jung Kyu Canci and I proved as a corollary that a rational function defined over with everywhere good reduction has at most periodic points. As a special case, we remarked that a monic polynomial with integer coefficients has at most -rational periodic points. It turns out that much better can be said, and that G. Baron proved in 1991 in an unpublished paper that such a polynomial has at most periodic points, including the point at infinity. I decided to translate the article from the original German, and bring it the mathematical community’s attention. I marked in red additions made by me. Any mistakes are probably due to my translation.
About integer polynomials with fixed points and two-cycles
Let be an element in the polynomial ring , where is some commutative ring. For an element one can define a sequence with and , for . A cycle is a finite sequence such that for and . The length of a cycle is . Over the ring we say that a cycle is integral if for any , and rational if for any .
W. Narkiewicz has shown that a polynomial has no [integral] cycle of length more than two and conjectured also that [integral] cycles of length two can occur only if there is no fixed point. [In particular, since monic polynomials in can only have integral cycles, one concludes that monic polynomials in have rational cycles of length at most .]
Question: Does there exist a polynomial , with a rational cycle of length ?
For the sake of completeness, we give an overview of the proof of the result of Narkiewicz.
Theorem 1 Let and , and define the sequence with and . Then is either (1) pairwise distinct, (2) eventually constant [i.e., = for all , therefore, cycles of length 1, which are fixed points] or (3) finally alternating [i.e., , , for all , i.e., two-cycles].
Proof: with . It follows by induction that , where the product runs over all from to .
- If for a pair , then is eventually periodic.
- In the special case and the sequence is eventually constant.
- If , then the following applies: where . So and for all relevant . If all , then , a contradiction. So a for some , and , and therefore and thus .
Corollary 2 A polynomial can only have [integral] cycles of length (two-cycles) and (fixed points).
Next, we want to show that both can occur simultaneously and thus refute the conjecture by Narkiewicz.
Theorem 3 Let be natural numbers with and pairwise distinct positive integers and a (monic) polynomial. The (monic) polynomial has the fixed point , and the two-cycles , .
Corollary 4 For each with there are monic polynomials in of degree with two-cycles and a fixed point.
Theorem 5 Let be natural numbers with and pairwise distinct integers other than zero and a (monic) polynomial. Then the (monic) polynomial has the fixed points and , .
Corollary 6 For each there are monic polynomials in of degree with fixed points.
Theorem 7 If a polynomial has an [integral] two-cycle, then has at most one fixed point.
Proof: Proof by contradiction. If has the two-cycle and two distinct fixed points (in particular ). Since , the polynomial satisfies , and thus there exists a polynomial such that . Therefore . From it follows and because also . From the division chain it therefore follows that and . Since and we get contradicting .
Theorem 8 If has two [integral] two-cycles and , then .
Proof: Since and then the polynomial satisfies . Therefore there exists a polynomial such that . So . The polynomial has the fixed points . Because of the shape of , we get
The polynomial has the roots . So . From it follows that and from it follows that . With we get . Thus, either both and are , or . From follows , and thus (remark that we only needed ). If we get , etc. From follows and , so , a contradiction (as before). From follows and therefore each one of two expressions equal to . But it follows again that , a contradiction.
Corollary 9 If has the [integral] two-cycles () with fixed such that for all , then is of the form where . is monic iff is monic.
Definition 10 Two polynomials are called equivalent [(over )] if there exists , so that .
Corollary 11 If is even, i.e., , then is equivalent to with the cycles , .
Proof: Take .
Theorem 12 If has the [integral] two-cycle and the fixed point , then .
Proof: Analogous to the proof of Theorem~8.
Corollary 13 If has the [integral] two-cycle with odd, then has no [integral] fixed point.
Exactly one of the following cases occurs: (where is monic exactly when is monic).
- Let have the fixed points , , then . If , then has no [integral] two-cycle.
- Let have the [integral] two-cycles , , then (fixed) and .
- If is even, then is equivalent to with the two-cycles , .
- If is odd, then is equivalent to with the two-cycles , .
- Let have a fixed point and [integral] two-cycles ,, so that , for , and .
- is equivalent to with the fixed point and the two-cycles .
Corollary 14 A polynomial of degree has at most integral periodic points. A monic polynomial of degree has at most rational periodic points.