Heuristics for the asymptotics of the number of S_n-number fields

2020/06/17 by Arul Shankar, Jacob Tsimerman, Shankar, Arul +1
Computer Science · Mathematics · #11R04 #11R09 #11R16 #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2006.09620

openalex publication_date 2020/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a heuristic argument supporting conjectures of Bhargava on the asymptotics of the number of S_n-number fields having bounded discriminant. We then make our arguments rigorous in the case n=3 giving a new elementary proof of the Davenport-Heilbronn theorem. Our basic method is to count elements of small height in S_n-fields while carefully keeping track of the index of the monogenic ring that they generate.

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