226 research outputs found
Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials
Let be a real closed field, with \deg_{Y}(Q) \leq 2, \deg_{X}(Q) \leq
d, Q \in {\mathcal Q}, #({\mathcal Q})=m, and with \deg_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal
P})=s. Let be a semi-algebraic set defined by a
Boolean formula without negations, with atoms . We describe an algorithm for computing the the
Betti numbers of . The complexity of the algorithm is bounded by . The complexity of the algorithm interpolates between the
doubly exponential time bounds for the known algorithms in the general case,
and the polynomial complexity in case of semi-algebraic sets defined by few
quadratic inequalities known previously. Moreover, for fixed and this
algorithm has polynomial time complexity in the remaining parameters.Comment: 24 pages, 3 figure
Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems
We give a survey of algorithms for computing topological invariants of
semi-algebraic sets with special emphasis on the more recent developments in
designing algorithms for computing the Betti numbers of semi-algebraic sets.
Aside from describing these results, we discuss briefly the background as well
as the importance of these problems, and also describe the main tools from
algorithmic semi-algebraic geometry, as well as algebraic topology, which make
these advances possible. We end with a list of open problems.Comment: Survey article, 74 pages, 15 figures. Final revision. This version
will appear in the AMS Contemporary Math. Series: Proceedings of the Summer
Research Conference on Discrete and Computational Geometry, Snowbird, Utah
(June, 2006). J.E. Goodman, J. Pach, R. Pollack Ed
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
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
Computing the homology of basic semialgebraic sets in weak exponential time
We describe and analyze an algorithm for computing the homology (Betti
numbers and torsion coefficients) of basic semialgebraic sets which works in
weak exponential time. That is, out of a set of exponentially small measure in
the space of data the cost of the algorithm is exponential in the size of the
data. All algorithms previously proposed for this problem have a complexity
which is doubly exponential (and this is so for almost all data)
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
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