23 research outputs found

    Computing the Real Isolated Points of an Algebraic Hypersurface

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    Let R\mathbb{R} be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in Rn\mathbb{R}^n 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 (nd)O(nlog(n))(nd)^{O(n\log(n))} for computing the real isolated points of real algebraic hypersurfaces of degree dd. It allows us to solve in practice instances which are out of reach of the state-of-the-art.Comment: Conference paper ISSAC 202

    Robots, computer algebra and eight connected components

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    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 88

    On the complexity of computing real radicals of polynomial systems

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    International audienceLet f= (f1, ..., fs) be a sequence of polynomials in Q[X1,...,Xn] of maximal degree D and V⊂ Cn be the algebraic set defined by f and r be its dimension. The real radical re associated to f is the largest ideal which defines the real trace of V . When V is smooth, we show that re , has a finite set of generators with degrees bounded by V. Moreover, we present a probabilistic algorithm of complexity (snDn )O(1) to compute the minimal primes of re . When V is not smooth, we give a probabilistic algorithm of complexity sO(1) (nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ Rn. Experiments are given to show the efficiency of our approaches

    On the Generation of Positivstellensatz Witnesses in Degenerate Cases

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    One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.Comment: To appear in ITP 201

    A baby steps/giant steps Monte Carlo algorithm for computing roadmaps in smooth compact real hypersurfaces

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    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 ss polynomial equations with rational coefficients, of degree DD in nn variables, Canny's algorithm has a Monte Carlo cost of snlog(s)DO(n2)s^n\log(s) D^{O(n^2)} operations in Q\mathbb{Q}; a deterministic version runs in time snlog(s)DO(n4)s^n \log(s) D^{O(n^4)}. The next improvement was due to Basu, Pollack and Roy, with an algorithm of deterministic cost sd+1DO(n2)s^{d+1} D^{O(n^2)} for the more general problem of computing roadmaps of semi-algebraic sets (dnd \le n is the dimension of an associated object). We give a Monte Carlo algorithm of complexity (nD)O(n1.5)(nD)^{O(n^{1.5})} for the problem of computing a roadmap of a compact hypersurface VV of degree DD in nn variables; we also have to assume that VV has a finite number of singular points. Even under these extra assumptions, no previous algorithm featured a cost better than DO(n2)D^{O(n^2)}
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