Polynomial Values in Subfields and Affine Subspaces of Finite Fields

TL;DR

This paper establishes bounds on how often polynomial iterates and affine subspace elements in finite fields fall into subfields or subspaces, revealing structural distribution properties of these elements.

Contribution

It provides new upper bounds on the frequency of polynomial orbit elements in subfields and affine subspaces of finite fields, advancing understanding of their distribution.

Findings

Upper bounds on polynomial orbit elements in subfields

Results on elements in affine subspaces as linear spaces

Insights into element distribution in finite fields

Abstract

For an integer $r$, a prime power $q$, and a polynomial $f$ over a finite field ${\mathbb F}_{q^r}$ of $q^r$ elements, we obtain an upper bound on the frequency of elements in an orbit generated by iterations of $f$ which fall in a proper subfield of ${\mathbb F}_{q^r}$. We also obtain similar results for elements in affine subspaces of ${\mathbb F}_{q^r}$, considered as a linear space over ${\mathbb F}_q$.