TY - JOUR

T1 - Squarefree values of polynomial discriminants I

AU - Bhargava, Manjul

AU - Shankar, Arul

AU - Wang, Xiaoheng

N1 - Funding Information:
We thank Levent Alpoge, Benedict Gross, Wei Ho, Kiran Kedlaya, Hendrik Lenstra, Barry Mazur, Bjorn Poonen, Peter Sarnak, and Ila Varma for their kind interest and many helpful conversations. The first and third authors were supported by a Simons Investigator Grant and NSF Grant DMS-1001828.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022

Y1 - 2022

N2 - We determine the density of monic integer polynomials of given degree n> 1 that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials f(x), such that f(x) is irreducible and Z[x] / (f(x)) is the ring of integers in its fraction field, is positive, and is in fact given by ζ(2) - 1. It also follows from our methods that there are ≫ X1/2+1/n monogenic number fields of degree n having associated Galois group Sn and absolute discriminant less than X, and we conjecture that the exponent in this lower bound is optimal.

AB - We determine the density of monic integer polynomials of given degree n> 1 that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials f(x), such that f(x) is irreducible and Z[x] / (f(x)) is the ring of integers in its fraction field, is positive, and is in fact given by ζ(2) - 1. It also follows from our methods that there are ≫ X1/2+1/n monogenic number fields of degree n having associated Galois group Sn and absolute discriminant less than X, and we conjecture that the exponent in this lower bound is optimal.

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U2 - 10.1007/s00222-022-01098-w

DO - 10.1007/s00222-022-01098-w

M3 - Article

AN - SCOPUS:85124261423

SN - 0020-9910

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

ER -