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Squarefree values of polynomial discriminants

Recent developments in Analytic Number Theory May 01, 2017 - May 05, 2017

May 02, 2017 (03:30 PM PDT - 04:30 PM PDT)
Speaker(s): Manjul Bhargava (Princeton University)
Location: MSRI: Simons Auditorium
  • sieve theory

  • Ekedahl Sieve

  • monogenic number fields

  • polynomial discriminants

  • geometry of numbers

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC



The question as to whether there exist a positive proportion of monic irreducible integral polynomials of degree n having squarefree discriminant is an old one; an exact formula for the density was conjectured by Lenstra.  (The interest in monic irreducible polynomials f with squarefree discriminant comes from the fact that in such cases

Z[x]/(f(x)) gives the ring of integers in the number field Q[x]/(f(x)).)


In this talk, we describe recent work with Arul Shankar and Xiaoheng Wang that allows us to determine the probability that a random monic integer polynomial has squarefree discriminant - thus proving the conjecture of Lenstra.


28395?type=thumb Bhargava Notes 1.24 MB application/pdf Download
Video/Audio Files


H.264 Video 7-Bhargava.mp4 229 MB video/mp4 rtsp://videos.msri.org/data/000/028/312/original/7-Bhargava.mp4 Download
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