Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants

Abstract

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.

Galois groups of random integer polynomials and van der Waerden's Conjecture

“…n (H) denotes the number of such polynomials having Galois group not equal to S n , then we may show similarly that E * n (H) = O(H n ) (improving on the previously best known estimate O ǫ (H n+ 1 3 + 8 9n+21 +ǫ ) due to [2]), thus proving van der Waerden's Conjecture also in this non-monic context (that the number of reducible polynomials gives the correct order of magnitude for the count of all polynomials having Galois group not equal to S n ). The analogues of Theorem 2 may be similarly developed for this family as well as for families of polynomials where some of the coefficients a i are fixed.…”

“…Zywina [47] (2010), who refined this to E n (H) = O(H n−1/2 ) for large n; Dietmann [14] (2012), who proved using resolvent polynomials and the determinant method that E n (H) = O(H n−2+ √ 2 ); and Anderson, Gafni, Lemke Oliver, Lowry-Duda, Shakan, and Zhang [2] (2021) who prove that E n (H) = O(H n− 2 3 + 2 3n+3 +ǫ ). For n ≤ 4, van der Waerden's conjectured optimal upper bound of O(H n−1 ) was proven by Chow and Dietmann [10].…”

On the number of integral binary n$n$‐ic forms having bounded Julia invariant

“…Specifically, van der Waerden showed that among the ∼ (2𝐻) 𝑛 monic integral polynomials of degree 𝑛 whose coefficients are bounded in absolute value by 𝐻, at most 𝑂(𝐻 𝑛−6∕((𝑛−2) log log 𝑛) ) have associated Galois group not 𝑆 𝑛 . This was subsequently improved by Gallagher [14] to 𝑂(𝐻 𝑛−1∕2 log 𝐻), by Zywina [28] to 𝑂(𝐻 𝑛−1∕2 ), by Dietmann [12] to 𝑂(𝐻 𝑛−2+ √ 2+𝜀 ), by Anderson, Gafni, Lemke Oliver, Lowry-Duda, Shakan, and Zhang [1] to 𝑂(𝐻 𝑛− 2 3 + 2 3𝑛+3 +𝜀 ), and most recently to the optimal 𝑂(𝐻 𝑛−1 ) in [3].…”

“…The invariant was constructed in terms of the discriminant of a certain canonical but irrational positive-definite SL 2 (ℝ)-covariant binary quadratic form 𝑄 of 𝑓. More precisely, consider a binary 𝑛-ic form 𝑓 with coefficients as in (1). If 𝑎 0 ≠ 0, we may write 𝑓(𝑥, 𝑦) = 𝑎 0 (𝑥 − 𝛼 1 𝑦)(𝑥 − 𝛼 2 𝑦) ⋯ (𝑥 − 𝛼 𝑛 𝑦) with 𝛼 𝑖 ∈ ℂ, and then, for any vector 𝑡 = (𝑡 1 , … , 𝑡 𝑛 ) of positive real numbers, we may consider the positive-definite quadratic form 𝑄 𝑡 (𝑥, 𝑦) = 𝑛 ∑ 𝑗=1 𝑡 2 𝑗 (𝑥 − 𝛼 𝑗 𝑦)(𝑥 − 𝛼 𝑗 𝑦).…”