Abstract. We discuss here the problem of the existence of polynomials with prescribed cycle lengths. From Šarkovskii's theorem we know that for every real polynomial $f$, the set of lengths of all cycles generated by $f$ is of the form $\mathop{\rm Cycl} (f)= \{m\in{\msbm N}\colon m\succeq n\} $ for some $n\in{\msbm N}$, where $\succeq $ denotes Šarkovskii's ordering. We show that for every odd integer $n\geq3$ there exists a polynomial $f$ such that $\mathop{\rm Cycl} (f)=\{m\in{\msbm N}\colon m\succeq n\} $. Moreover, our proof works not only over ${\msbm R}$ but over any real closed field.

AMS Subject Classification (1991): 12D15, 11C08, 26A18, 39B12

Keyword(s): polynomial cycles, iterations, cycle lengths, Šarkovskii's theorem, real closed fields

Received October 22, 2004, and in final form February 1, 2006. (Registered under 5906/2009.)