Towards van der Waerden’s conjecture

Chow, Sam; Dietmann, Rainer

doi:10.1090/tran/8784

Cited by 3 publications

(1 citation statement)

References 45 publications

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“…In 2021, Bhargava [5] made a breakthrough by establishing a weak version of the van der Waerden conjecture:

\textnormal {Prob}(G_f\ne S_n) = O(L^{-1})

. See also the two preceding results [1, 12] in the same year. In all of the above, the most challenging part is to bound the probability that

G_f=A_n

.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Galois Theory: The square discriminant case

Bary‐Soroker,

Ben‐Porath,

Matei

2024

Bulletin of London Math Soc

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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.

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“…In 2021, Bhargava [5] made a breakthrough by establishing a weak version of the van der Waerden conjecture:

\textnormal {Prob}(G_f\ne S_n) = O(L^{-1})

. See also the two preceding results [1, 12] in the same year. In all of the above, the most challenging part is to bound the probability that

G_f=A_n

.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic Galois Theory: The square discriminant case

Bary‐Soroker,

Ben‐Porath,

Matei

2024

Bulletin of London Math Soc

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Galois groups of random additive polynomials

Bary-Soroker,

Entin,

McKemmie

2024

Trans. Amer. Math. Soc.

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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.

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Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

Anderson

Gafni

Oliver

et al. 2021

Preprint

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We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert's Irreducibility Theorem for degree n polynomials f with Galpf q Ď An. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree n monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree n number fields with almost prime discriminants.

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Towards van der Waerden’s conjecture

Cited by 3 publications

References 45 publications

Probabilistic Galois Theory: The square discriminant case

Probabilistic Galois Theory: The square discriminant case

Galois groups of random additive polynomials

Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

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