Computing the multiplicity structure in solving polynomial systems

Dayton, Barry H.; Zeng, Zhonggang

doi:10.1145/1073884.1073902

Cited by 99 publications

(169 citation statements)

References 20 publications

Supporting

Mentioning

164

Contrasting

Unclassified

Order By: Relevance

“…When a mechanism has several assembly modes with different degrees of freedom, the applicable mathematical terminology is local dimension. The local dimension test described here is based on prior work [1] using multiplicity ideas from [3,6]. Taken alone, the rank tests cannot distinguish between a finite degree of freedom and a high-multiplicity infinitesimal degree of freedom for which the Macaulay corank sequence stabilizes at a depth greater than the prespecified depth bound.…”

Section: Discussionmentioning

confidence: 99%

“…The relation between vanishing derivatives, multiplicity, and dimension is easiest to understand through a construction we call a Macaulay matrix [18], recently reintroduced in a modern computational context by Dayton and Zeng [6]. This matrix organizes the analysis of higher derivatives.…”

Section: Multiplicity and Local Dimensionmentioning

confidence: 99%

“…. , f m (x)) has columns indexed by α and rows indexed by (β, j) with elements In some presentations of the Macaulay matrix, e.g., [6], the elements are defined as…”

Section: Multiplicity and Local Dimensionmentioning

confidence: 99%

See 2 more Smart Citations

Mechanism mobility and a local dimension test

Wampler

Hauenstein

Sommese

2011

Mechanism and Machine Theory

View full text Add to dashboard Cite

The mobility of a mechanism is the number of degrees of freedom (DOF) with which it may move. This notion is mathematically equivalent to the dimension of the solution set of the kinematic loop equations for the mechanism. It is well known that the classical Grübler-Kutzbach formulas for mobility can be wrong for special classes of mechanisms, and even more refined treatments based on displacement groups fail to correctly predict the mobility of so-called "paradoxical" mechanisms. This article discusses how recent results from numerical algebraic geometry can be applied to the question of mechanism mobility. In particular, given an assembly configuration of a mechanism and its loop equations, a local dimension test places bounds on the mobility of the associated assembly mode. A publicly available software code makes the idea easy to apply in the kinematics domain.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Multiplicity and Local Dimensionmentioning

confidence: 99%

See 1 more Smart Citation

Mechanism mobility and a local dimension test

Wampler

Hauenstein

Sommese

2011

Mechanism and Machine Theory

View full text Add to dashboard Cite

show abstract

“…Deflation for isolated solutions was introduced in [OWM83,Oji87] with a proof of termination provided in [LVZ06] (see also [DZ05,LVZ08,HSW10]). Deflation was extended from isolated solutions to irreducible components in [SW05, §13.3.2, §15.2.2].…”

Section: A Numerical Algorithm For Computing the Components Of An Affmentioning

confidence: 99%

Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

Bates

Decker

Hauenstein

et al. 2014

Applied Mathematics and Computation

View full text Add to dashboard Cite

Systems of polynomial equations arise throughout mathematics, engineering, and the sciences. It is therefore a fundamental problem both in mathematics and in application areas to find the solution sets of polynomial systems. The focus of this paper is to compare two fundamentally different approaches to computing and representing the solutions of polynomial systems: numerical homotopy continuation and symbolic computation. Several illustrative examples are considered, using the software packages Bertini and Singular.

show abstract

“…Letting {F (x), G(x, λ)} and (x * , λ * ) play the roles of the original F and x * , one may repeat the construction iteratively until one arrives at a system that has a nonsingular root corresponding to x * . Dayton and Zeng [5] used a construction called a Macaulay matrix to help understand the number of iterations required to reach nonsingularity, and this lead to a higherorder deflation method in [13] and the closedness-subspace method [33]. (On another track, a symbolic deflation method was presented in [12].)…”

Section: Deflation and Isosingular Setsmentioning

confidence: 99%

Numerically intersecting algebraic varieties via witness sets

Hauenstein¹,

Wampler²

2013

Applied Mathematics and Computation

View full text Add to dashboard Cite

The fundamental construct of numerical algebraic geometry is the representation of an irreducible algebraic set, A, by a witness set, which consists of a polynomial system, F , for which A is an irreducible component of V(F ), a generic linear space L of complementary dimension to A, and a numerical approximation to the set of witness points, L ∩ A. Given F , methods exist for computing a numerical irreducible decomposition, which consists of a collection of witness sets, one for each irreducible component of V(F ). This paper concerns the more refined question of finding a numerical irreducible decomposition of the intersection A ∩ B of two irreducible algebraic sets, A and B, given a witness set for each. An existing algorithm, the diagonal homotopy, computes witness point supersets for A ∩ B, but this does not complete the numerical irreducible decomposition. In this paper, we use the theory of isosingular sets to complete the process of computing the numerical irreducible decomposition of the intersection.

show abstract

Computing the multiplicity structure in solving polynomial systems

Cited by 99 publications

References 20 publications

Mechanism mobility and a local dimension test

Mechanism mobility and a local dimension test

Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

Numerically intersecting algebraic varieties via witness sets

Contact Info

Product

Resources

About