The orbit intersection problem in positive characteristic

TL;DR

This paper investigates the orbit intersection problem for affine and algebraic group maps over fields of positive characteristic, establishing conditions under which the intersection sets are structured as p-normal sets or finite unions of simple families.

Contribution

It provides new results characterizing orbit intersections in positive characteristic, including p-normality and finiteness results under eigenvalue and polynomial conditions.

Findings

The set of intersection points is p-normal of order at most d in the affine case.

Under certain conditions, the intersection set is a finite union of points and linear families in the multiplicative group case.

Uses results on polynomial-exponential equations and linear equations over multiplicative groups in positive characteristic.

Abstract

In this paper, we study the orbit intersection problem for the linear space and the algebraic group in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic and let $\Phi_1, \Phi_{2}: K^d \longrightarrow K^{d}$ 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 none of the eigenvalues of the matrices $A_i$ are roots of unity and each ${\bf a}_i \in K^d$ is not $\Phi_i$-preperiodic, then we prove that the set $\left \{(n_1, n_2) \in \Z^{2} \mid \Phi_1^{n_1}({\bf a}_1) = \Phi_{2}^{n_{2}}({\bf a}_{2})\right\}$ is $p$-normal in $\mathbb{Z}^{2}$ of order at most $d$. Further, let $\Phi_1, \Phi_{2}: \mathbb{G}_m^d \longrightarrow \mathbb{G}_m^d$ be regular self-maps and ${\bf a}_1, {\bf a}_2\in \mathbb{G}_m^d(K)$. Let $\Phi_1^0$ and $\Phi_2^0$ be group endomorphisms of $\mathbb{G}_m^d$ and ${\bf y}, {\bf z} \in \mathbb{G}_m^d(K)$ 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 polynomial of $ \Phi_1^{0}$ and $\Phi_2^{0}$, that the set $ \{(n_1, n_{2}) \in \N_0^{2} \mid \Phi_1^{n_1}({\bf a}_1) = \Phi_{2}^{n_{2}}({\bf a}_{2})\}$ (where ${\bf a}_1, {\bf a}_2\in \mathbb{G}_m^d(K)$) 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.