Enumerative Galois theory for cubics and quartics

Abstract: This is joint work with Sam Chow. We consider monic quartic polynomials with integer coefficients and growing box height at most H. In this setting, we exactly determine the order of magnitude (from above and below) of such polynomials whose Galois group is D_4. Moreover, we show that C_4 and V_4 polynomials are less frequent that D_4 ones, and that D_4, C_4, V_4 and A_4 polynomials are together less frequent than reducible quartics. Similarly, for integer monic cubic polynomials we show that A_3 cubics are less frequent than reducible cubics. In particular, irreducible non-S_n polynomials are less numerous than reducible ones for n = 3 and n = 4, for the first time solving two cases (namely degree three and four) of a conjecture by van der Waerden from 1936.

number theory

Audience: researchers in the topic

Tel Aviv field arithmetic seminar

Series comments: Please sign in to the talk with a full name.

The link to the seminar will be given before the talk in the seminar website www.math.tau.ac.il/~haran/Field-Arithmetic-Seminar/