Irreducibility of Random Polynomials

Abstract

ABSTRACTWe 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 support 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 indicate 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 1. 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 nonlinear, low-degree factor. We also study characteristic polynomials of random matrices with + 1 and − 1 entries.