Irreducibility of Random Polynomials

2017/05/10 by Borst, Christian, Boyd, Evan, Brekken, Claire +3
#11C08 #60C05 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.1705.03709

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.

Related

• On the probability of irreducibility of random polynomials with integer coefficients
• Low-degree factors of random polynomials
• Irreducibility of Polynomials with Square Coefficients over Finite Fields
• Irreducibility of random polynomials of ℤ[X]
• A random polynomial with multiplicative coefficients is almost surely irreducible