8,113 research outputs found
An improved bound on the number of point-surface incidences in three dimensions
We show that points and smooth algebraic surfaces of bounded degree
in satisfying suitable nondegeneracy conditions can have at most
incidences, provided that any
collection of points have at most O(1) surfaces passing through all of
them, for some . In the case where the surfaces are spheres and no
three spheres meet in a common circle, this implies there are point-sphere incidences. This is a slight improvement over the previous
bound of for an (explicit) very
slowly growing function. We obtain this bound by using the discrete polynomial
ham sandwich theorem to cut into open cells adapted to the set
of points, and within each cell of the decomposition we apply a Turan-type
theorem to obtain crude control on the number of point-surface incidences. We
then perform a second polynomial ham sandwich decomposition on the irreducible
components of the variety defined by the first decomposition. As an
application, we obtain a new bound on the maximum number of unit distances
amongst points in .Comment: 17 pages, revised based on referee comment
A Polyhedral Homotopy Algorithm For Real Zeros
We design a homotopy continuation algorithm, that is based on numerically
tracking Viro's patchworking method, for finding real zeros of sparse
polynomial systems. The algorithm is targeted for polynomial systems with
coefficients satisfying certain concavity conditions. It operates entirely over
the real numbers and tracks the optimal number of solution paths. In more
technical terms; we design an algorithm that correctly counts and finds the
real zeros of polynomial systems that are located in the unbounded components
of the complement of the underlying A-discriminant amoeba.Comment: some cosmetic changes are done and a couple of typos are fixed to
improve readability, mathematical contents remain unchange
Enumerative Real Algebraic Geometry
Enumerative Geometry is concerned with the number of solutions to a
structured system of polynomial equations, when the structure comes from
geometry. Enumerative real algebraic geometry studies real solutions to such
systems, particularly a priori information on their number. Recent results in
this area have, often as not, uncovered new and unexpected phenomena, and it is
far from clear what to expect in general. Nevertheless, some themes are
emerging.
This comprehensive article describe the current state of knowledge,
indicating these themes, and suggests lines of future research. In particular,
it compares the state of knowledge in Enumerative Real Algebraic Geometry with
what is known about real solutions to systems of sparse polynomials.Comment: Revised, corrected version. 40 pages, 18 color .eps figures. Expanded
web-based version at http://www.math.umass.edu/~sottile/pages/ERAG/index.htm
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
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