Enumerative Galois theory for cubics and quartics

We show that there are Oε(H1.5+ε) monic, cubic polynomials with integer coefficients bounded by H in absolute value whose Galois group is A3. We also show that the order of magnitude for D4 quartics is H2(log⁡H)2, and that the respective counts for A4, V4, C4 are O(H2.91), O(H2log⁡H), O(H2log⁡H). Ou...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2020-10, Vol.372, p.107282, Article 107282
Main Authors: Chow, Sam, Dietmann, Rainer
Format: Article
Language:English
Subjects:
Determinant method
Galois theory
Mahler measure
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Summary:We show that there are Oε(H1.5+ε) monic, cubic polynomials with integer coefficients bounded by H in absolute value whose Galois group is A3. We also show that the order of magnitude for D4 quartics is H2(log⁡H)2, and that the respective counts for A4, V4, C4 are O(H2.91), O(H2log⁡H), O(H2log⁡H). Our work establishes that irreducible non-S3 cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2020.107282