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DTSTART:20260308T070000
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DTSTART;TZID=America/Toronto:20260407T100000
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URL:https://uwaterloo.ca/pure-mathematics/events/number-theory-seminar-165
SUMMARY:Number Theory Seminar
CLASS:PUBLIC
DESCRIPTION:ILA VARMA\, UNIVERSITY OF TORONTO\n\n_Counting Number Fields by
  P^1 height_\n\nWhen do two irreducible polynomials with integer coefficie
 nts define\nthe same number field? One can define an action of GL_2 × GL_
 1 on the\nspace of polynomials of degree n so that for any two polynomials
  f and\ng in the same orbit\, the roots of f may be expressed as rational\
 nlinear transformations of the roots of g\; thus\, they generate the same\
 nfield. In this article\, we show that almost all polynomials of degree\nn
  with size at most X can only define the same number field as another\npol
 ynomial of degree n with size at most X if they lie in the same\norbit for
  this group action. (Here we measure the size of polynomials\nby the great
 est absolute value of their coefficients.)\n\nThis improves on work of Bha
 rgava\, Shankar\, and Wang\, who proved a\nsimilar statement for a positiv
 e proportion of polynomials. Using this\nresult\, we prove that the number
  of degree n fields such that the\nsmallest polynomial defining the field 
 has size at most X is\nasymptotic to a constant times X^{n+1} as long as n
  \\ge 3. For n = 2\,\nwe obtain a precise asymptotic of the form 27/(pi^2)
  * X^2\n\nThis is joint work with Arango-Pineros\, Gundlach\, Lemke Olive
 r\,\nMcGown\, Sawin\, Serrano Lopez\, and Shankar.\n\nMC 5479
DTSTAMP:20260406T180617Z
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