# Cycles of polynomials with integer coefficients

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 ${\phi:\mathbb{P}^1\rightarrow\mathbb{P}^1}$ defined over ${\mathbb{Q}}$ with everywhere good reduction has at most ${d+5}$ periodic points. As a special case, we remarked that a monic polynomial with integer coefficients has at most ${d+5}$ ${\mathbb{Q}}$-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 It turns out that much better can be said, and that G. Baron proved in 1991 in an unpublished paper that such a polynomial has at most ${d+1}$ periodic points, including the point at infinity.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.