In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.

Many polynomial systems have solution sets comprising multiple irreducible components, possibly of dierent dimensions. A fundamental problem of numerical algebraic geometry is to decompose such a solution set, using oating-point numerical processes, into its components.

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Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We describe a numerically stable algorithm for decomposing positive dimensional solution sets of polynomial systems into irreducible components.

Abstract. We show how to use numerical continuation to compute the intersection C = A∩B of two algebraic sets A and B, where A, B, and C are numerically represented by witness sets. Enroute to this result, we first show how to find the irreducible decomposition of a system of polynomials restricted to an algebraic set. The intersection of components A and B then follows by considering the decomposition of the diagonal system of equations u − v = 0 restricted to {u, v} ∈ A × B. One offshoot of this new approach is that one can solve a large system of equations by finding the solution components of its subsystems and then intersecting these. It also allows one to find the intersection of two components of the two polynomial systems, which is not possible with any previous numerical continuation approach.

To classify positive dimensional solution components of a polynomial system, we construct polynomials interpolating points sampled from each component. In previous work, points on an i-dimensional component were linearly projected onto a generically chosen (i + 1)-dimensional subspace. In this paper, we present two improvements. First, we reduce the dimensionality of the ambient space by determining the linear span of the component and restricting to it. Second, if the dimension of the linear span is greater than i + 1, we use a less generic projection that leads to interpolating polynomials of lower degree, thus reducing the number of samples needed. While this more ecient approach still guarantees | with probability one | the correct determination of the degree of each component, the mere evaluation of an interpolating polynomial no longer certi es the membership of a point to that component. We present an additional numerical test that certi es membership in this new situation. We illustrate the performance of our new approach on some well-known test systems.

One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment.

For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higher-dimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.

In the numerical treatment of solution sets of polynomial systems, methods for sampling and tracking a path on a solution component are fundamental. For example, in the numerical irreducible decomposition of a solution set for a polynomial system, one first obtains a "witness point set" containing generic points on all the irreducible components and then these points are grouped via numerical exploration of the components by path tracking from these points. A numerical difficulty arises when a component has multiplicity greater than one, because then all points on the component are singular. This paper overcomes this di#culty using an embedding of the polynomial system in a family of systems such that in the neighborhood of the original system each point on a higher multiplicity solution component is approached by a cluster of nonsingular points. In the case of the numerical irreducible decomposition, this embedding can be the same embedding that one uses to generate the witness point set. In handling the case of higher multiplicities, this paper, in concert with the methods we previously proposed to decompose reduced solution components, provides a complete algorithm for the numerical irreducible decomposition. The method is applicable to tracking singular paths in other contexts as well.