A slight improvement to Schmidt's bound for "Number fields of given degree and bounded discriminant"

Kevin Hughes (Edinburgh Napier University)

Wednesday 21st February 16:00-17:00 Maths 110


For degree n, at least 2, it is conjectured that the number of number fields (over the rationals) of degree n whose discriminant lies in [-X,X] grows asymptotically as c_n X as X goes to infinity for some constant depending on the degree. This is known to be true for degrees 2,3,4 and 5, but is wide open for degrees strictly greater than 5. I will discuss recent work with T. Anderson, A. Gafni, R. Lemke Oliver, D. Lowry-Duda, F. Thorne, J. Wang and R. Zhang on this problem. Using a variety of analytic techniques, we obtain a humble improvement of Schmidt's upper bound from 1995 in the range of degrees between 6 and 94 inclusive. For higher degrees better bounds were previously known.

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