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تسجيل مستخدم جديدDynatomic polynomials, necklace operators, and universal relations for dynamical units
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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 $f(x)$.
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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.
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