Irreducibility of random polynomials: general measures

Bary-Soroker, Lior; Koukoulopoulos, Dimitris; Kozma, Gady

doi:10.1007/s00222-023-01193-6

Cited by 4 publications

(1 citation statement)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we study the so‐called large box model , in which we fix the degree

n=\deg f

f

and we choose the coefficients uniformly and independently from

[-L,L]\cap \mathbb {Z}

, and we are interested in the probabilities as

L

goes to

\infty

. See [3, 4, 8] for recent progress on the restricted coefficient model in which

L

is fixed and

n\rightarrow \infty

and [15, 16] for recent progress in the random matrix model (where

f

is the characteristic polynomial of a random matrix).…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Galois Theory: The square discriminant case

Bary‐Soroker,

Ben‐Porath,

Matei

2024

Bulletin of London Math Soc

View full text Add to dashboard Cite

The paper studies the probability for a Galois group of a random polynomial to be . We focus on the so‐called large box model, where we choose the coefficients of the polynomial independently and uniformly from . The state‐of‐the‐art upper bound is , due to Bhargava. We conjecture a much stronger upper bound , and that this bound is essentially sharp. We prove strong lower bounds both on this probability and on the related probability of the discriminant being a square.

show abstract

“…In this paper, we study the so‐called large box model , in which we fix the degree

n=\deg f

f

and we choose the coefficients uniformly and independently from

[-L,L]\cap \mathbb {Z}

, and we are interested in the probabilities as

L

goes to

\infty

. See [3, 4, 8] for recent progress on the restricted coefficient model in which

L

is fixed and

n\rightarrow \infty

and [15, 16] for recent progress in the random matrix model (where

f

is the characteristic polynomial of a random matrix).…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Galois Theory: The square discriminant case

Bary‐Soroker,

Ben‐Porath,

Matei

2024

Bulletin of London Math Soc

View full text Add to dashboard Cite

show abstract

Probabilistic Galois theory in function fields

Entin,

Popov

2024

Finite Fields and Their Applications

View full text Add to dashboard Cite

Galois groups of random additive polynomials

Bary-Soroker,

Entin,

McKemmie

2024

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We study the distribution of the Galois group of a random q q -additive polynomial over a rational function field: For q q a power of a prime p p , let f = X q n + a n − 1 X q n − 1 + … + a 1 X q + a 0 X f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots +a_1X^q+a_0X be a random polynomial chosen uniformly from the set of q q -additive polynomials of degree n n and height d d , that is, the coefficients are independent uniform polynomials of degree deg ⁡ a i ≤ d \deg a_i\leq d . The Galois group G f G_f is a random subgroup of GL n ⁡ ( q ) \operatorname {GL}_n(q) . Our main result shows that G f G_f is almost surely large as d , q d,q are fixed and n → ∞ n\to \infty . For example, we give necessary and sufficient conditions so that SL n ⁡ ( q ) ≤ G f \operatorname {SL}_n(q)\leq G_f asymptotically almost surely. Our proof uses the classification of maximal subgroups of GL n ⁡ ( q ) \operatorname {GL}_n(q) . We also consider the limits: q , n q,n fixed, d → ∞ d\to \infty and d , n d,n fixed, q → ∞ q\to \infty , which are more elementary.

show abstract

Irreducibility of random polynomials: general measures

Cited by 4 publications

References 29 publications

Probabilistic Galois Theory: The square discriminant case

Probabilistic Galois Theory: The square discriminant case

Probabilistic Galois theory in function fields

Galois groups of random additive polynomials

Contact Info

Product

Resources

About