An Algorithm for Determining the Dynamical Irreducibility of Cubic Polynomials over Finite Fields

(Last updated: 12/13/2025)

Table of Contents

• Description
• What you can find here
• Using the Algorithm
• Reflections
• Acknowledgement
• Reference

Description

Tech used: Magma

This algorithm was developed as part of the research project on Dynamical Irreducibiilty of Polynomials over Finite Fields at Mount Holyoke College, done under the mentorship of Professor Tori Day.

It is used for determining if a cubic polynomial of the form f(x) = ax^3 + bx^2 + cx + d is dynamically irreducible over a given finite field of order q (q being a prime or prime power), and is based on a Sage algorithm for determining the dynamically irreducibility of polynomials of the form f(x) = ax^3 + cx + d (Day, Deland, Juul, Thomas, Thompson, Tobin).

What you can find here

Here is a brief guide for navigating through the repository.

• In the algorithm_versions folder, you will find the code for the algorithm (Algorithm2_v2) and its helper functions.
• The archive folder contains the older versions of the algorithm. (will be updated soon)
• In the test folder, you will find some functions for performing automated tests on the algorithm.

Using the Algorithm

If you haven't installed Magma on your computer, here is a guide for that: Installation Instructions

If you have installed Magma, follow these steps to use the algorithm:

1. Download the files Helper_Functions.txt, Algorithm1_v4.txt and Algorithm2_v3.txt in the folder algorithm_versions.
2. Open Magma, then load the previously downloaded files into Magma in this order: Helper_Functions.txt -> Algorithm1_v4.txt -> Algorithm2_v3.txt. You can do that using this Magma command:

   load "filepath"; // paste the filepath for the file you want to load in between the quotation marks
   

The algorithm should now be ready to use! You can test it out on creating a polynomial of your choice, like so:

Example:

// Create polynomial over finite field of order 7
F:=FiniteField(7);
P<x>:=PolynomialRing(F);
f:=x^3+2*x^2+x+1;

// Check if polynomial f is dynamically irreducible over 5 iterations
Algorithm2(f, 7, 5, 1);

Reflections

Through this project, I was involved in math research for the first time - which led to a series of other first-times: learning Magma, presenting about out research at MHC math lunch, and finding out how fun math can be!

The research process was a great learning opportunity. Through weeks of reading theories, coding, testing, debugging, I had the chance to expand my comfort zone, step by step.

• I practiced breaking down into pieces a problem that seemed daunting at first sight, and solving it piece by piece.
• I found two new tools for tackling problems: composure and confidence. A lot of times, bugs that were confusing at first can be fixed by letting myself take time, and believing there was a solution to be found.
• I learned to voice my questions and concerns - and was truly grateful to be among great people who are willing to give me support and guidance ☺️

Acknowledgement

I would like to extend my gratitude to:

• Mount Holyoke College's Math/Stat/Data Department - for supporting this summer research project.

• Professors Tony Liu and Chassidy Bozeman, whom I took Data Structures and Discrete Math with. The coding and proofwriting skills I learned from your classes proved useful in this project!

• My Kachimushi seniors, Pemily, Viktoria and Gráinne. Thanks to you all I had something fun to look forward to every week (I might not look like I was having fun at practice, but I really was).

• My friend and trusted proofreader Định, for proofreading this README.

• My mentor, Professor Tori Day, for your guidance throughout this project. I learned a lot from your knowledgeableness, enthusiasm and patience.

Reference

1. Tori Day, Rebecca Deland, Jamie Juul, Cigole Thomas, Bianca Thompson, and Bella Tobin, "Dynamical Irreducibility of Certain Families of Polynomials over Finite Fields" (2024). https://arxiv.org/abs/2409.10467.