1,463 research outputs found
Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial
In this thesis, we consider semi-algebraic sets over a real closed field
defined by quadratic polynomials. Semi-algebraic sets of are defined as
the smallest family of sets in that contains the algebraic sets as well
as the sets defined by polynomial inequalities, and which is also closed under
the boolean operations (complementation, finite unions and finite
intersections). We prove new bounds on the Betti numbers as well as on the
number of different stable homotopy types of certain fibers of semi-algebraic
sets over a real closed field defined by quadratic polynomials, in terms of
the parameters of the system of polynomials defining them, which improve the
known results. We conclude the thesis with presenting two new algorithms along
with their implementations. The first algorithm computes the number of
connected components and the first Betti number of a semi-algebraic set defined
by compact objects in which are simply connected. This algorithm
improves the well-know method using a triangulation of the semi-algebraic set.
Moreover, the algorithm has been efficiently implemented which was not possible
before. The second algorithm computes efficiently the real intersection of
three quadratic surfaces in using a semi-numerical approach.Comment: PhD thesis, final version, 109 pages, 9 figure
Combinatorial complexity in o-minimal geometry
In this paper we prove tight bounds on the combinatorial and topological
complexity of sets defined in terms of definable sets belonging to some
fixed definable family of sets in an o-minimal structure. This generalizes the
combinatorial parts of similar bounds known in the case of semi-algebraic and
semi-Pfaffian sets, and as a result vastly increases the applicability of
results on combinatorial and topological complexity of arrangements studied in
discrete and computational geometry. As a sample application, we extend a
Ramsey-type theorem due to Alon et al., originally proved for semi-algebraic
sets of fixed description complexity to this more general setting.Comment: 25 pages. Revised version. To appear in the Proc. London Math. So
Computing the First Few Betti Numbers of Semi-algebraic Sets in Single Exponential Time
In this paper we describe an algorithm that takes as input a description of a
semi-algebraic set , defined by a Boolean formula with atoms of
the form for
and outputs the first Betti numbers of ,
The complexity of the algorithm is where where s =
#({\mathcal P}) and which is
singly exponential in for any fixed constant. Previously, singly
exponential time algorithms were known only for computing the Euler-Poincar\'e
characteristic, the zero-th and the first Betti numbers
On the number of homotopy types of fibres of a definable map
In this paper we prove a single exponential upper bound on the number of
possible homotopy types of the fibres of a Pfaffian map, in terms of the format
of its graph. In particular we show that if a semi-algebraic set , where is a real closed field, is defined by a Boolean formula
with polynomials of degrees less than , and
is the projection on a subspace, then the number of different homotopy types of
fibres of does not exceed . As applications
of our main results we prove single exponential bounds on the number of
homotopy types of semi-algebraic sets defined by fewnomials, and by polynomials
with bounded additive complexity. We also prove single exponential upper bounds
on the radii of balls guaranteeing local contractibility for semi-algebraic
sets defined by polynomials with integer coefficients.Comment: Improved combinatorial complexit
Approximation of definable sets by compact families, and upper bounds on homotopy and homology
We prove new upper bounds on homotopy and homology groups of o-minimal sets
in terms of their approximations by compact o-minimal sets. In particular, we
improve the known upper bounds on Betti numbers of semialgebraic sets defined
by quantifier-free formulae, and obtain for the first time a singly exponential
bound on Betti numbers of sub-Pfaffian sets.Comment: 20 pages, 2 figure
Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities
We prove a nearly optimal bound on the number of stable homotopy types
occurring in a k-parameter semi-algebraic family of sets in , each
defined in terms of m quadratic inequalities. Our bound is exponential in k and
m, but polynomial in . More precisely, we prove the following. Let
be a real closed field and let with . Let be a
semi-algebraic set, defined by a Boolean formula without negations, whose atoms
are of the form, . Let be the projection on the last k co-ordinates. Then, the number of
stable homotopy types amongst the fibers S_{\x} = \pi^{-1}(\x) \cap S is
bounded by Comment: 27 pages, 1 figur
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