#### Vol. 12, No. 7, 2018

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A dynamical variant of the Pink–Zilber conjecture

### Dragos Ghioca and Khoa Dang Nguyen

Vol. 12 (2018), No. 7, 1749–1771
##### Abstract

Let ${f}_{1},\dots ,{f}_{n}\in \overline{ℚ}\left[x\right]$ be polynomials of degree $d>1$ such that no ${f}_{i}$ is conjugate to ${x}^{d}$ or to $±{C}_{d}\left(x\right)$, where ${C}_{d}\left(x\right)$ is the Chebyshev polynomial of degree $d$. We let $\phi$ be their coordinatewise action on ${\mathbb{A}}^{n}$, i.e., $\phi :{\mathbb{A}}^{n}\to {\mathbb{A}}^{n}$ is given by $\left({x}_{1},\dots ,{x}_{n}\right)↦\left({f}_{1}\left({x}_{1}\right),\dots ,{f}_{n}\left({x}_{n}\right)\right)$. We prove a dynamical version of the Pink–Zilber conjecture for subvarieties $V$ of ${\mathbb{A}}^{n}$ with respect to the dynamical system $\left({\mathbb{A}}^{n},\phi \right)$, if $min\left\{dim\left(V\right),codim\left(V\right)-1\right\}\le 1$.

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