# The Chebotarev density theorem for function fields -- incomplete intervals

## TL;DR

This paper extends the Chebotarev density theorem to incomplete intervals in function fields over finite fields, providing new density results for irreducible polynomials and their factorization types.

## Contribution

It proves a Polya-Vinogradov type variation of the Chebotarev density theorem for incomplete intervals, with applications to irreducible trinomials and polynomial factorization distributions.

## Key findings

- Density of irreducible trinomials approximates expected value under certain interval size conditions.
- Distribution of polynomial factorization types matches cycle type distribution in symmetric groups.
- Applicable to polynomials with degree d over finite fields, generalizing previous results.

## Abstract

We prove a Polya-Vinogradov type variation of the the Chebotarev density theorem for function fields over finite fields valid for "incomplete intervals" $I \subset \mathbb{F}_p$, provided $(p^{1/2}\log p)/|I| = o(1)$. Applications include density results for irreducible trinomials in $\mathbb{F}_p[x]$, i.e. the number of irreducible polynomials in the set $\{ f(x) = x^{d} + a_{1} x + a_{0} \in \mathbb{F}_p[x] \}_{a_{0} \in I_{0}, a_{1}\in I_{1}}$ is $\sim |I_{0}|\cdot |I_{1}|/d$ provided $|I_{0}| > p^{1/2+\epsilon}$, $|I_{1}| > p^{\epsilon}$, or $|I_{1}| > p^{1/2+\epsilon}$, $|I_{0}| > p^{\epsilon}$, and similarly when $x^{d}$ is replaced by any monic degree $d$ polynomial in $\mathbb{F}_p[x]$. Under the above assumptions we can also determine the distribution of factorization types, and find it to be consistent with the distribution of cycle types of permutations in the symmetric group $S_{d}$.