The orbit intersection problem in positive characteristic
Volume 206 / 2022
Abstract
We study the orbit intersection problem for a linear space and an algebraic group in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic and let $\Phi _1, \Phi _{2}: K^d \rightarrow K^{d}$ be affine maps, $\Phi _i({\bf x}) = A_i ({\bf x}) + {\bf x_i}$ (where each $A_i$ is a $d\times d$ matrix and ${\bf x}\in K^d$). If no eigenvalue of $A_i$ is a root of unity and no ${\bf a}_i \in K^d$ is $\Phi _i$-preperiodic, then the set $\{(n_1, n_2) \in \mathbb Z ^{2} \mid \Phi _1^{n_1}({\bf a}_1) = \Phi _{2}^{n_{2}}({\bf a}_{2})\}$ is $p$-normal in $\mathbb {Z}^{2}$ of order at most $d$. Further, let $\Phi _1, \Phi _{2}: \mathbb G_{\rm m}^d \rightarrow \mathbb G_{\rm m}^d$ be regular self-maps and ${\bf a}_1, {\bf a}_2\in \mathbb G_{\rm m}^d(K)$. Let $\Phi _1^0$ and $\Phi _2^0$ be group endomorphisms of $\mathbb G_{\rm m}^d$ and ${\bf y}, {\bf z} \in \mathbb G_{\rm m}^d(K)$ be such that $\Phi _1({\bf x}) = \Phi _1^{0}({\bf x}) + {\bf y}$ and $\Phi _2({\bf x}) = \Phi _2^{0}({\bf x}) + {\bf z}$. We show, under some conditions on the roots of the minimal polynomials of $ \Phi _1^{0}$ and $\Phi _2^{0}$, that the set $ \{(n_1, n_{2}) \in \mathbb N _0^{2} \mid \Phi _1^{n_1}({\bf a}_1) = \Phi _{2}^{n_{2}}({\bf a}_{2})\}$ (where ${\bf a}_1, {\bf a}_2\in \mathbb G_{\rm m}^d(K)$) is a finite union of singletons and one-parameter linear families. To do so, we use results on linear equations over multiplicative groups in positive characteristic and some results on systems of polynomial-exponential equations.
