6 research outputs found
Local Box Adjacency Algorithms for Cylindrical Algebraic Decompositions
AbstractWe describe new algorithms for determining the adjacencies between zero-dimensional cells and those one-dimensional cells that are sections (not sectors) in cylindrical algebraic decompositions (cad). Such adjacencies constitute a basis for determining all other cell adjacencies. Our new algorithms are local, being applicable to a specified 0D cell and the 1D cells described by specified polynomials. Particularly efficient algorithms are given for the 0D cells in spaces of dimensions two, three and four. Then an algorithm is given for a space of arbitrary dimension. This algorithm may on occasion report failure, but it can then be repeated with a modified isolating interval and a likelihood of success
Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems
We consider the problem of counting (stable) equilibriums of an important
family of algebraic differential equations modeling multistable biological
regulatory systems. The problem can be solved, in principle, using real
quantifier elimination algorithms, in particular real root classification
algorithms. However, it is well known that they can handle only very small
cases due to the enormous computing time requirements. In this paper, we
present a special algorithm which is much more efficient than the general
methods. Its efficiency comes from the exploitation of certain interesting
structures of the family of differential equations.Comment: 24 pages, 5 algorithms, 10 figure
Exact geometric-topological analysis of algebraic surfaces
We present a method to compute the exact topology of a real algebraic surface
, implicitly given by a polynomial of arbitrary
degree .
Additionally, our analysis provides geometric information as it
supports the computation of arbitrary precise samples of
including critical points.
We use a projection approach,
similar to Collins' cylindrical algebraic decomposition (cad).
In comparison we reduce the number of output cells to by constructing
a special planar arrangement instead of a full cad in the projection plane.
Furthermore, our approach applies numerical and combinatorial methods to
minimize costly symbolic computations. The algorithm handles all sorts of
degeneracies without transforming the surface into a generic position.
We provide a complete implementation of the algorithm, written in C++. It shows
good performance for many well known examples from algebraic geometry
Isotopic triangulation of a real algebraic surface
International audienceWe present a new algorithm for computing the topology of a real algebraic surface in a ball , even in singular cases. We use algorithms for 2D and 3D algebraic curves and show how one can compute a topological complex equivalent to , and even a simplicial complex isotopic to by exploiting properties of the contour curve of . The correctness proof of the algorithm is based on results from stratification theory. We construct an explicit Whitney stratification of , by resultant computation. Using Thom's isotopy lemma, we show how to deduce the topology of from a finite number of characteristic points on the surface. An analysis of the complexity of the algorithm and effectiveness issues conclude the paper
On the isotopic meshing of an algebraic implicit surface
International audienceWe present a new and complete algorithm for computing the topology of an algebraic surface given by a squarefree polynomial in Q[X, Y, Z]. Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in [15], on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface S. We exploit simple algebraic criterion to certify the pseudo-genericity and genericity position of the surface. This gives us rational parametrizations of the components of the polar curve, which are used to lift the topology of the projection of the polar curve. We deduce the connection of the two-dimensional components above the cell defined by the projection of the polar curve. A complexity analysis of the algorithm is provided leading to a bound in OB (d15 τ ) for the complexity of the computation of the topology of an implicit algebraic surface defined by integer coefficients polynomial of degree d and coefficients size τ . Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces