125 research outputs found
An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions
We prove an asymptotically tight bound (asymptotic with respect to the number
of polynomials for fixed degrees and number of variables) on the number of
semi-algebraically connected components of the realizations of all realizable
sign conditions of a family of real polynomials. More precisely, we prove that
the number of semi-algebraically connected components of the realizations of
all realizable sign conditions of a family of polynomials in
whose degrees are at most is bounded by This improves the best upper bound known
previously which was The new
bound matches asymptotically the lower bound obtained for families of
polynomials each of which is a product of generic polynomials of degree one.Comment: 19 pages. Bibliography has been updated and a few more references
have been added. This is the final version of this paper which will appear in
Combinatoric
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
Recommended from our members
Reelle Algebraische Geometrie
This workshop was organized by Michel Coste (Rennes), Claus Scheiderer (Konstanz) and Niels Schwartz (Passau). The talks focussed on recent developments in real enumerative and tropical geometry, positivity and sums of squares, real aspects of classical algebraic geometry, semialgebraic and tame geometry, and topology and singularities of real varieties
Pseudo-Triangulations, Rigidity and Motion Planning
This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc
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Geometric, Algebraic, and Topological Combinatorics
The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz
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