16 research outputs found
On the geometry of polar varieties
We have developed in the past several algorithms with intrinsic complexity
bounds for the problem of point finding in real algebraic varieties. Our aim
here is to give a comprehensive presentation of the geometrical tools which are
necessary to prove the correctness and complexity estimates of these
algorithms. Our results form also the geometrical main ingredients for the
computational treatment of singular hypersurfaces.
In particular, we show the non--emptiness of suitable generic dual polar
varieties of (possibly singular) real varieties, show that generic polar
varieties may become singular at smooth points of the original variety and
exhibit a sufficient criterion when this is not the case. Further, we introduce
the new concept of meagerly generic polar varieties and give a degree estimate
for them in terms of the degrees of generic polar varieties.
The statements are illustrated by examples and a computer experiment
Polar degrees and closest points in codimension two
Suppose that is a toric variety of codimension
two defined by an integer matrix , and let be a Gale
dual of . In this paper we compute the Euclidean distance degree and polar
degrees of (along with other associated invariants) combinatorially
working from the matrix . Our approach allows for the consideration of
examples that would be impractical using algebraic or geometric methods. It
also yields considerably simpler computational formulas for these invariants,
allowing much larger examples to be computed much more quickly than the
analogous combinatorial methods using the matrix in the codimension two
case.Comment: 25 pages, 1 figur
On the Complexity of the Generalized MinRank Problem
We study the complexity of solving the \emph{generalized MinRank problem},
i.e. computing the set of points where the evaluation of a polynomial matrix
has rank at most . A natural algebraic representation of this problem gives
rise to a \emph{determinantal ideal}: the ideal generated by all minors of size
of the matrix. We give new complexity bounds for solving this problem
using Gr\"obner bases algorithms under genericity assumptions on the input
matrix. In particular, these complexity bounds allow us to identify families of
generalized MinRank problems for which the arithmetic complexity of the solving
process is polynomial in the number of solutions. We also provide an algorithm
to compute a rational parametrization of the variety of a 0-dimensional and
radical system of bi-degree . We show that its complexity can be bounded
by using the complexity bounds for the generalized MinRank problem.Comment: 29 page
Computing the dimension of real algebraic sets
Let be the set of real common solutions to in
and be the maximum total degree of the
's. We design an algorithm which on input computes the dimension of
. Letting be the evaluation complexity of and , it runs using
arithmetic operations in and
at most isolations of real roots of polynomials of degree at most
. Our algorithm depends on the real geometry of ; its practical
behavior is more governed by the number of topology changes in the fibers of
some well-chosen maps. Hence, the above worst-case bounds are rarely reached in
practice, the factor being in general much lower on practical
examples. We report on an implementation showing its ability to solve problems
which were out of reach of the state-of-the-art implementations.Comment: v2: title chang
Robots, computer algebra and eight connected components
Answering connectivity queries in semi-algebraic sets is a long-standing and
challenging computational issue with applications in robotics, in particular
for the analysis of kinematic singularities. One task there is to compute the
number of connected components of the complementary of the singularities of the
kinematic map. Another task is to design a continuous path joining two given
points lying in the same connected component of such a set. In this paper, we
push forward the current capabilities of computer algebra to obtain
computer-aided proofs of the analysis of the kinematic singularities of various
robots used in industry. We first show how to combine mathematical reasoning
with easy symbolic computations to study the kinematic singularities of an
infinite family (depending on paramaters) modelled by the UR-series produced by
the company ``Universal Robots''. Next, we compute roadmaps (which are curves
used to answer connectivity queries) for this family of robots. We design an
algorithm for ``solving'' positive dimensional polynomial system depending on
parameters. The meaning of solving here means partitioning the parameter's
space into semi-algebraic components over which the number of connected
components of the semi-algebraic set defined by the input system is invariant.
Practical experiments confirm our computer-aided proof and show that such an
algorithm can already be used to analyze the kinematic singularities of the
UR-series family. The number of connected components of the complementary of
the kinematic singularities of generic robots in this family is