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Given a generic polynomial $f(x)$, the generalized dynatomic polynomial $Phi_{f,c,d}(x)$ vanishes at precisely those $alpha$ such that $f^c(alpha)$ has period exactly $d$ under iteration of $f(x)$. We show that the shifted dynatomic polynomials $Phi_{f,c,d}(x) - 1$ often have generalized dynatomic factors, and that these factors are in correspondence with certain cyclotomic factors of necklace polynomials. These dynatomic factors of $Phi_{f,c,d}(x) - 1$ have an interpretation in terms of new multiplicative relations between dynamical units which are uniform in the polynomial $f(x)$.
Let $f in Q(z)$ be a polynomial or rational function of degree 2. A special case of Morton and Silvermans Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of $f$ is bounded above by an absolute constant.
We give necessary and sufficient conditions for post-critically finite polynomials to have potential good reduction at a given prime. We also answer in the negative a question posed by Silverman about conservative polynomials. Both proofs rely on dyn
Given an endomorphism of a projective variety, by intersecting the graph and the diagonal varieties we can determine the set of periodic points. In an effort to determine the periodic points of a given minimal period, we follow a construction similar
Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height hat{h}_f is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of f. Morton and Silverman conjectured in 1994 that the
Let $K$ be the function field of a smooth, irreducible curve defined over $overline{mathbb{Q}}$. Let $fin K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r ge 1,$ is a power of the prime number $p$, and let $betain overline{K}$. For all $ninmathb