Squarefree values of polynomial discriminants I

• 成果类型: Article
• 署名作者: Bhargava, Manjul; Shankar, Arul; Wang, Xiaoheng
• 署名单位: Princeton University; University of Toronto; University of Waterloo
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01098-w
• 发表日期: 2022
• 页码: 1037-1073
• 关键词: hyperelliptic curves; number; EXTENSIONS; field

摘要

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 zeta (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 S-n and absolute discriminant less than X, and we conjecture that the exponent in this lower bound is optimal.