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Title: 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 $\textrm {Gal}(f) \subseteq A_n$. 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.  more » « less
Award ID(s):
2231990 2207281
NSF-PAR ID:
10406227
Author(s) / Creator(s):
; ; ; ; ;
Date Published:
Journal Name:
International Mathematics Research Notices
Volume:
2023
Issue:
3
ISSN:
1073-7928
Page Range / eLocation ID:
2188 to 2214
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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