Gyeonghyeon Nam | Searcharxiv

arXiv2024

Intersection of orbits for polynomials in characteristic $p$

In [GTZ08, GTZ12], the following result was established: given polynomials $f,g\in\mathbb{C}[x]$ of degrees larger than $1$, if there exist $α,β\in\mathbb{C}$ such that their corresponding orbits $\mathcal{O}_f(α)$ and $\mathcal{O}_g(β)$ (under the action of $f$, respectively of $g$) intersect in infinitely many points, then $f$ and $g$ must share a common iterate, i.e., $f^m=g^n$ for some $m,n\in\mathbb{N}$. If one replaces $\mathbb{C}$ with a field $K$ of characteristic $p$, then the conclusion fails; we provide numerous examples showing the complexity of the problem over a field of positive characteristic. We advance a modified conjecture regarding polynomials $f$ and $g$ which admit two orbits with infinite intersection over a field of characteristic $p$. Then we present various partial results, along with connections with another deep conjecture in the area, the dynamical Mordell-Lang conjecture.

math.NT↗

arXiv2024

A converse of dynamical Mordell--Lang conjecture in positive characteristic

In this paper, we prove the converse of the dynamical Mordell--Lang conjecture in positive characteristic: For every subset $S \subseteq \mathbb{N}_0$ which is a union of finitely many arithmetic progressions along with finitely many $p$-sets of the form $\left \{ \sum_{j=1}^{m} c_j p^{k_jn_j} : n_j \in \mathbb{N}_0 \right \}$ ($c_j \in \mathbb{Q}$, $k_j \in \mathbb{N}_0$), there exist a split torus $X = \mathbb{G}_m^k$ defined over $K=\overline{\mathbb{F}_p}(t)$, an endomorphism $Φ$ of $X$, $α\in X(K)$ and a closed subvariety $V \subseteq X$ such that $\left \{ n \in \mathbb{N}_0 : Φ^n(α) \in V(K) \right \} = S$.

math.NT↗