Irreducibility of Random Polynomials

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.We study many polynomial models that appear to satisfy Heuristic 1.1 (primarily with independent coefficients, though some with dependence, see Section 4), and the "well-behaved" condition is meant to exclude models with specific features that cause high-degree factors, for example, a random polynomial of degree d formed as the product of two random polynomials with degree around d/2. Throughout, "reducible" will be used as shorthand for "reducible over the rationals." Note that for monic polynomials, reducibility over the rationals is equivalent to reducibility over the integers by Gauss's Lemma. For non-monic polynomials, we are interested in cases where all Key words and phrases. random integer polynomials, irreducible, low-degree factors.

Unlocking the walk matrix of a graph

“…In the original context, see also [20], an eigenvector is said to be a main eigenvector if its eigenspace contains a vector that is not perpendicular to e. This definition is equivalent to the one we are using here: Lemma 3. 3 The eigenvalue μ i is a main eigenvalue for e if and only if Eig(A, μ i ) contains a vector that is not perpendicular to e. Proof Clearly, E i e belongs to Eig(A, μ i ), and if E i e = 0 then e T (E i e) = e T (E 2 i e) = (E i e) T (E i e) = 0 by Lemma 2.2. Conversely, suppose that E i a is not perpendicular to e for some a ∈ K n .…”

“…In Theorem 5.2 we have noted that if char G (x) is irreducible then W S is invertible for any vertex set S. Therefore, in this case the conclusion of Theorem 7.1 holds for walk matrices of general type. In the literature there are several papers in which this irreducibility problem is considered from a probabilistic point of view, see [3,4,8,19,26]. In fact, there is the The following two adjacency matrices on 8 vertices…”

“…In particular, when S is set of vertices then its characteristic vector e can be written in this way. For convenience, we renumber the distinct eigenvalues of A so that SD(S) : e = e 1 + e 2 + • • • + e r with e i = 0 and Ae i = μ i e i for all 1 ≤ i ≤ r (3) for some r ≤ s. We refer to (3) as the spectral decomposition of S, or more properly, of its characteristic vector. We outline the main results.…”

“…The key theorem shows that the walk matrix for S determines the spectral decomposition of S, and vice versa. This holds for any graph G and any non-empty set S of vertices of G. To state this in a precise fashion, we use (3) to define the n × r main eigenvector matrix E S = e 1 , e 2 , . .…”

Irreducibility of random polynomials of large degree

“…In another paper, Bary-Soroker and Kozma [4] studied the problem for bivariate polynomials. See also [31] for a study of the probability that a random polynomial has low degree factors, and [6] for computational experiments on related problems.…”