Irreducibility of random polynomials of bounded degree

  title={Irreducibility of random polynomials of bounded degree},
  author={Huy-Tuan Pham and Max Wenqiang Xu},
  journal={arXiv: Number Theory},
  • H. PhamM. Xu
  • Published 24 February 2020
  • Mathematics
  • arXiv: Number Theory
It is known that random monic integral polynomials of bounded degree $d$ and integral coefficients distributed uniformly and independently in $[-H,H]$ are irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$. In this paper, we prove that the same conclusion holds under much more general distributions of the coefficients, allowing them to be dependently and nonuniformly distributed. 

Irreducibility and Galois Groups of Random Polynomials

Let f(x) be a random integral polynomial of degree d ≥ 2 with coefficients uniformly and independently drawn from [−N,N ]. It is well known that the probability that f(x) is irreducible over the

Towards van der Waerden’s conjecture

  • Sam ChowR. Dietmann
  • Computer Science, Art
    Transactions of the American Mathematical Society
  • 2023
It is established that the number of such polynomials, monic and irreducible with integer coefficients in <inline-formula content-type="math/mathml" xmlns:mml="" alttext="n greater-than-or-slanted-equals 3") is established.



Irreducible polynomials of bounded height

The goal of this paper is to prove that a random polynomial with i.i.d. random coefficients taking values uniformly in $\{1,\ldots, 210\}$ is irreducible with probability tending to $1$ as the degree

Irreducibility of random polynomials of large degree

We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are

Galois Groups of Generic Polynomials

We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{\Omega(d)}N}{N}.$

Irreducibility of Random Polynomials

The data support conjectures made by Odlyzko and Poonen and by Konyagin, and a universality heuristic and new conjectures that connect their work with Hilbert’s Irreducibility Theorem and work of van der Waerden are formulated.

Low-Degree Factors of Random Polynomials

This work shows that pointwise delocalization of the roots of a random polynomials can be used to imply that the polynomial is unlikely to have a low-degree factor over the integers.

Enumerative Galois theory for quartics

We show that there are order of magnitude $H^2 (\log H)^2$ monic quartic polynomials with integer coefficients having box height at most $H$ whose Galois group is $D_4$. Further, we prove that the

Distribution of reducible polynomials with a given coefficient set

For a given set of integers $\mathcal{S}$, let $\mathcal{R}_n^*(\mathcal{S})$ denote the set of reducible polynomials $f(X)=a_nX^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ over $\mathbb{Z}[X]$ with

Corrections to: “The Galois group of a polynomial with two indeterminate coefficients”

Φ 0) is a polynomial in which two of the coefficients are indeterminates t, u and the remainder belong to a field F. We find the galois group of / over F(t, u). In particular, it is the full

Heights in Diophantine Geometry

I. Heights II. Weil heights III. Linear tori IV. Small points V. The unit equation VI. Roth's theorem VII. The subspace theorem VIII. Abelian varieties IX. Neron-Tate heights X. The Mordell-Weil