40 research outputs found
Probabilistic Galois Theory
We show that there are at most
monic integer polynomials of degree having height at most and Galois
group different from the full symmetric group , improving on the previous
1973 world record .Comment: 10 page
Enumerative Galois theory for cubics and quartics
We show that there are monic, cubic
polynomials with integer coefficients bounded by in absolute value whose
Galois group is . We also show that the order of magnitude for
quartics is , and that the respective counts for , ,
are , , . Our work establishes
that irreducible non- 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
Rational lines on cubic hypersurfaces
We show that any smooth projective cubic hypersurface of dimension at least
over the rationals contains a rational line. A variation of our methods
provides a similar result over p-adic fields. In both cases, we improve on
previous results due to the second author and Wooley.
We include an appendix in which we highlight some slight modifications to a
recent result of Papanikolopoulos and Siksek. It follows that the set of
rational points on smooth projective cubic hypersurfaces of dimension at least
29 is generated via secant and tangent constructions from just a single point.Comment: An oversight in Lemma 3.1 as well as a few typos have been correcte