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

# About integer polynomials with fixed points and two-cycles

### Gerd Baron

Let *cycle* is a finite sequence *length* of a cycle *integral* if *rational* if

W. Narkiewicz has shown that a polynomial *rational* cycles of length at most

**Question:** Does there exist a polynomial

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:*

- 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

**Theorem 8** * If has two [integral] two-cycles and , then . *

*Proof:* Since

The polynomial

**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

- 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 , .

- If
- 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. *