859 research outputs found
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
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
Mathematical Abstraction, Conceptual Variation and Identity
One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject
Mathematical Abstraction, Conceptual Variation and Identity
One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject
The Utility of Naturalness, and how its Application to Quantum Electrodynamics envisages the Standard Model and Higgs Boson
With the Higgs boson discovery and no new physics found at the LHC,
confidence in Naturalness as a guiding principle for particle physics is under
increased pressure. We wait to see if it proves its mettle in the LHC upgrades
ahead, and beyond. In the meantime, in a series of "realistic intellectual
leaps" I present a justification {\it a posteriori} of the Naturalness
criterion by suggesting that uncompromising application of the principle to
quantum electrodynamics leads toward the Standard Model and Higgs boson without
additional experimental input. Potential lessons for today and future theory
building are commented upon.Comment: 7 pages, no figure
Partition regularity and multiplicatively syndetic sets
We show how multiplicatively syndetic sets can be used in the study of
partition regularity of dilation invariant systems of polynomial equations. In
particular, we prove that a dilation invariant system of polynomial equations
is partition regular if and only if it has a solution inside every
multiplicatively syndetic set. We also adapt the methods of Green-Tao and
Chow-Lindqvist-Prendiville to develop a syndetic version of Roth's density
increment strategy. This argument is then used to obtain bounds on the Rado
numbers of configurations of the form .Comment: 29 pages. v3. Referee comments incorporated, accepted for publication
in Acta Arithmetic
Roots of polynomials of degrees 3 and 4
We present the solutions of equations of degrees 3 and 4 using Galois theory
and some simple Fourier analysis for finite groups, together with historical
comments on these and other solution methods.Comment: 29 page
- …