708 research outputs found
Regulous vector bundles
Among recently introduced new notions in real algebraic geometry is that of
regulous functions. Such functions form a foundation for the development of
regulous geometry. Several interesting results on regulous varieties and
regulous sheaves are already available. In this paper, we define and
investigate regulous vector bundles. We establish algebraic and geometric
properties of such vector bundles, and identify them with stratified-algebraic
vector bundles. Furthermore, using new results on curve-rational functions, we
characterize regulous vector bundles among families of vector spaces
parametrized by an affine regulous variety. We also study relationships between
regulous and topological vector bundles
Global variants of Hartogs’ theorem
Hartogs’ theorem asserts that a separately holomorphic function, defined on an open subset of Cn, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions
Solving moment problems by dimensional extension
The first part of this paper is devoted to an analysis of moment problems in
R^n with supports contained in a closed set defined by finitely many polynomial
inequalities. The second part of the paper uses the representation results of
positive functionals on certain spaces of rational functions developed in the
first part, for decomposing a polynomial which is positive on such a
semi-algebraic set into a canonical sum of squares of rational functions times
explicit multipliers.Comment: 21 pages, published version, abstract added in migratio
Singularities of free group character varieties
Let X be the moduli space of SL(n,C), SU(n), GL(n,C), or U(n)-valued
representations of a rank r free group. We classify the algebraic singular
stratification of X. This comes down to showing that the singular locus
corresponds exactly to reducible representations if there exist singularities
at all. Then by relating algebraic singularities to topological singularities,
we show the moduli spaces X generally are not topological manifolds, except for
a few examples we explicitly describe.Comment: 33 pages. Version 4 is shorter and more focused; cut material will be
expanded upon and written up in subsequent papers. Clarifications, and
expository revisions have been added. Accepted for publication in Pacific
Journal of Mathematic
Non-Archimedean Whitney stratifications
We define "t-stratifications", a strong notion of stratifications for
Henselian valued fields of equi-characteristic 0, and prove that they
exist. In contrast to classical stratifications in Archimedean fields,
t-stratifications also contain non-local information about the stratified sets.
For example, they do not only see the singularities in the valued field, but
also those in the residue field.
Like Whitney stratifications, t-stratifications exist for different classes
of subsets of , e.g. algebraic subvarieties or certain classes of analytic
subsets. The general framework are definable sets (in the sense of model
theory) in a language that satisfies certain hypotheses.
We give two applications. First, we show that t-stratifications in suitable
valued fields induce classical Whitney stratifications in or ; in particular, the existence of t-stratifications implies the existence of
Whitney stratifications. This uses methods of non-standard analysis.
Second, we show how, using t-stratifications, one can determine the
ultra-metric isometry type of definable subsets of for
sufficiently big. For those , this proves a conjecture stated in a previous
article. In particular, this yields a new, geometric proof of the rationality
of Poincar\'e series.Comment: Fixed typos; enhanced the presentatio
L'anglais chat et les langues romanes
L’ANGLAIS chat ET LES LANGUES ROMANE
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