47 research outputs found
What is good mathematics?
Some personal thoughts and opinions on what ``good quality mathematics'' is,
and whether one should try to define this term rigorously. As a case study, the
story of Szemer\'edi's theorem is presented.Comment: 12 pages, no figures. To appear, Bull. Amer. Math. So
Independence in computable algebra
We give a sufficient condition for an algebraic structure to have a
computable presentation with a computable basis and a computable presentation
with no computable basis. We apply the condition to differentially closed, real
closed, and difference closed fields with the relevant notions of independence.
To cover these classes of structures we introduce a new technique of safe
extensions that was not necessary for the previously known results of this
kind. We will then apply our techniques to derive new corollaries on the number
of computable presentations of these structures. The condition also implies
classical and new results on vector spaces, algebraically closed fields,
torsion-free abelian groups and Archimedean ordered abelian groups.Comment: 24 page
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
Mathematical Logic: Proof Theory, Constructive Mathematics
[no abstract available
Degree spectra for transcendence in fields
We show that for both the unary relation of transcendence and the finitary
relation of algebraic independence on a field, the degree spectra of these
relations may consist of any single computably enumerable Turing degree, or of
those c.e. degrees above an arbitrary fixed degree. In other
cases, these spectra may be characterized by the ability to enumerate an
arbitrary set. This is the first proof that a computable field can
fail to have a computable copy with a computable transcendence basis
Infinite monochromatic patterns in the integers
We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials; in particular, we obtain extensions of both the additive and multiplicative versions of Hindman's theorem. These configurations are obtained by means of suitable symmetric polynomials that mix the two operations. The simplest example is the following. For every finite coloring N=C1∪…∪Cr there exists an infinite increasing sequence a<… such that all elements below are monochromatic: a,b,c,…,a+b+ab,a+c+ac,b+c+bc,…,a+b+c+ab+ac+bc+abc,…. The proofs use tools from algebra in the space of ultrafilters βZ