25 research outputs found
Computing the Real Isolated Points of an Algebraic Hypersurface
Let be the field of real numbers. We consider the problem of
computing the real isolated points of a real algebraic set in
given as the vanishing set of a polynomial system. This problem plays an
important role for studying rigidity properties of mechanism in material
designs. In this paper, we design an algorithm which solves this problem. It is
based on the computations of critical points as well as roadmaps for answering
connectivity queries in real algebraic sets. This leads to a probabilistic
algorithm of complexity for computing the real isolated
points of real algebraic hypersurfaces of degree . It allows us to solve in
practice instances which are out of reach of the state-of-the-art.Comment: Conference paper ISSAC 202
A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces
International audienceWe consider the problem of constructing roadmaps of real algebraic sets. The problem was introduced by Canny to answer connectivity questions and solve motion planning problems. Given polynomial equations with rational coefficients, of degree in variables, Canny's algorithm has a Monte Carlo cost of operations in ; a deterministic version runs in time . The next improvement was due to Basu, Pollack and Roy, with an algorithm of deterministic cost for the more general problem of computing roadmaps of semi-algebraic sets ( is the dimension of an associated object). We give a Monte Carlo algorithm of complexity for the problem of computing a roadmap of a compact hypersurface of degree in variables; we also have to assume that has a finite number of singular points. Even under these extra assumptions, no previous algorithm featured a cost better than
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
Critical Point Methods and Effective Real Algebraic Geometry: New Results and Trends
International audienceCritical point methods are at the core of the interplay between polynomial optimization and polynomial system solving over the reals. These methods are used in algorithms for solving various problems such as deciding the existence of real solutions of polynomial systems, performing one-block real quantifier elimination, computing the real dimension of the solution set, etc. The input consists of polynomials in variables of degree at most . Usually, the complexity of the algorithms is where is a constant. In the past decade, tremendous efforts have been deployed to improve the exponents in the complexity bounds. This led to efficient implementations and new geometric procedures for solving polynomial systems over the reals that exploit properties of critical points. In this talk, we present an overview of these techniques and their impact on practical algorithms. Also, we show how we can tune them to exploit algebraic and geometric structures in two fundamental problems. The first one is real root finding of determinants of -variate linear matrices of size . We introduce an algorithm whose complexity is polynomial in (joint work with S. Naldi and D. Henrion). This improves the previously known bound. The second one is about computing the real dimension of a semi-algebraic set. We present a probabilistic algorithm with complexity , that improves the long-standing bound obtained by Koi\-ran (joint work with E. Tsigaridas)