Low-Degree Factors of Random Polynomials

Abstract

We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is equivalent to having a low-degree algebraic root. It is known in certain cases that random polynomials with integer coefficients are very likely to be irreducible, and our project can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random polynomial models. Our main result shows that pointwise delocalization of the roots of a random polynomial can be used to imply that the polynomial is unlikely to have a low-degree factor over the integers. We apply our main result to a number of models of random polynomials, including characteristic polynomials of random matrices, where strong delocalization results are known.

Irreducibility of Random Polynomials

“…Proof. By a geometric series argument similar to [17,Lemma 4.1], x + 1 is the only possible linear factor for a polynomial g {0,1},d (x), and to have x + 1 as a factor, −1 must be a root. We will consider the cases of d odd and d even separately.…”

“…Even-degree polynomials g ±1,d (x) cannot have a linear factor, but these polynomials may still support a variant of Konyagin's conjecture where having a quadratic or other low-degree factor appears to be the most common way for such a polynomial to factor-see Lemma 5.2. Also, O'Rourke and Wood [17] prove that all polynomials of this form have a vanishingly small probability of having a factor with any fixed degree, including linear factors, as d goes to infinity. Data for both polynomials g {0,1},d (x) and g ±1,d (x) supports these conjectures and our Heuristic 1.1 (see Section 2).…”

“…O'Rourke and Wood [17] study a variety of models, proving that pointwise delocalization of the roots of a random polynomial implies that low-degree factors are very unlikely. Konyagin [12] studied random monic polynomials with 0 or 1 for coefficients and proved a bound on the probability of low-degree factors which he then used to prove a bound on the probability of reducibility.…”

“…Random polynomials with coefficients 1 or −1 have also been studied by Peled, Sen, and Zeitouni [18], and they prove that the probability of a double root is asymptotically equal to the probability Figure 3. The plot above shows probability of reducibility for monic polynomials where each coefficient is +1 or −1 independently with probability 1/2 (note that this figure also appears in [17,Figure 2]). The random data was calculated using 150,000,000 trials, giving a 99.9999% confidence interval of 0.0002 for every value.…”

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.…”