322 research outputs found
Topology of definable Hausdorff limits
Let be a set definable in an o-minimal expansion of the
real field, be its projection, and assume that the non-empty
fibers are compact for all and uniformly bounded,
{\em i.e.} all fibers are contained in a ball of fixed radius If
is the Hausdorff limit of a sequence of fibers we give an
upper-bound for the Betti numbers in terms of definable sets
explicitly constructed from a fiber In particular, this allows to
establish effective complexity bounds in the semialgebraic case and in the
Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative
closure} to construct the o-minimal structure \S_\pfaff generated by Pfaffian
functions in a way that is adapted to complexity problems. Our results can be
used to estimate the Betti numbers of a relative closure in the
special case where is empty.Comment: Latex, 23 pages, no figures. v2: Many changes in the exposition and
notations in an attempt to be clearer, references adde
Topological complexity of the relative closure of a semi-Pfaffian couple
Gabrielov introduced the notion of relative closure of a Pfaffian couple as
an alternative construction of the o-minimal structure generated by
Khovanskii's Pfaffian functions. In this paper, use the notion of format (or
complexity) of a Pfaffian couple to derive explicit upper-bounds for the
homology of its relative closure.
Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti
numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve
Quantitative study of semi-Pfaffian sets
We study the topological complexity of sets defined using Khovanskii's
Pfaffian functions, in terms of an appropriate notion of format for those sets.
We consider semi- and sub-Pfaffian sets, but more generally any definable set
in the o-minimal structure generated by the Pfaffian functions, using the
construction of that structure via Gabrielov's notion of limit sets. All the
results revolve around giving effective upper-bounds on the Betti numbers (for
the singular homology) of those sets.
Keywords: Pfaffian functions, fewnomials, o-minimal structures, tame
topology, spectral sequences, Morse theory.Comment: Author's PhD thesis. Approx. 130 pages, no figure
Relative Pfaffian closure for definably complete Baire structures
Speissegger proved that the Pfaffian closure of an o-
minimal expansion of the real field is o-minimal. Here we give a
first order version of this result: having introduced the notion of
definably complete Baire structure, we define the relative Pfaf-
fian closure of an o-minimal structure inside a definably complete
Baire structure, and we prove its o-minimality. We derive effec-
tive bounds on some topological invariants of sets definable in
the Pfaffian closure of an o-minimal expansion of the real field
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