# Unlikely intersections over finite fields: polynomial orbits in small subgroups

## TL;DR

This paper investigates how often polynomial iterations land in small subgroups of finite fields and provides bounds on subgroup sizes generated by polynomial orbits, advancing understanding of polynomial dynamics over finite fields.

## Contribution

It introduces new bounds on polynomial orbit intersections with subgroups and extends results to sequences of polynomial compositions over finite fields.

## Key findings

- Estimates the frequency of polynomial orbit intersections with subgroups.
- Provides lower bounds on subgroup sizes generated by polynomial orbits.
- Develops general results for sequences of polynomial compositions.

## Abstract

We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$ elements in an orbit. We derive these from more general results about sequences of compositions or a fixed set of polynomials.