# Irreducibility of Random Polynomials

## TL;DR

This paper investigates the likelihood of reducibility of random integer polynomials, supporting existing conjectures and proposing new heuristics, with extensive computational data across various models.

## Contribution

It provides computational evidence supporting conjectures on polynomial reducibility and introduces a universality heuristic linking these results with classical theorems.

## Key findings

- Probability of reducibility divided by linear factor probability approaches a constant.
- In large-degree limit, this constant tends to one.
- When linear factors are impossible, reducibility correlates with low-degree factors.

## Abstract

We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic polynomials. Our data supports conjectures made by Odlyzko and Poonen and by Konyagin, and we formulate a universality heuristic and new conjectures that connect their work with Hilbert's Irreducibility Theorem and work of van der Waerden. The data indicates that the probability that a random polynomial is reducible divided by the probability that there is a linear factor appears to approach a constant and, in the large-degree limit, this constant appears to approach one. In cases where the model makes it impossible for the random polynomial to have a linear factor, the probability of reducibility appears to be close to the probability of having a non-linear, low-degree factor. We also study characteristic polynomials of random matrices with +1 and -1 entries.