Seminar Algebra and Geometry: Youssef Fares (Amiens)

Some remarks on Poonen's conjecture

Let c be a rational number and consider the polynomial map φ_c(x)=x²-c.
We are interested in cycles of φ_c in $\Q$. More precisely, we focus on Poonen's conjecture, according to which every cycle of φ_c in $\Q$ is of length at most 3. In our talk, we discuss this conjecture using arithmetic, combinatorial and analytic means. In particular, we obtain an upper bound of the cardinality of the set of periodic points which we improve in the case c≤2. We finish the talk by giving some properties regarding rational numbers c for which φ_c has a cycle of length ≥ 4.

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