Homotopies for Connected Components of Algebraic Sets with Application to Computing Critical Sets

Bates, Daniel J.; Brake, Daniel A.; Hauenstein, Jonathan D.; Sommese, Andrew J.; Wampler, Charles W.

doi:10.1007/978-3-319-72453-9_8

Cited by 6 publications

(8 citation statements)

References 23 publications

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“…Moreover, using the observation in [2] further helps to reduce the computation by only using slices in x rather than slices in (x, λ). Such an approach can also be combined with intersection via regeneration [14] to compute the rank-deficiency set.…”

Section: Rank-deficiency Setsmentioning

confidence: 98%

“…It is well-known that F has two solutions for general parameters with generic Hilbert function [1,2,2] and index of regularity k * = 1. In particular, there is a linear relationship between c1 and s1.…”

Section: Example Using Rr Dyadmentioning

confidence: 99%

“…We now use the approach of Section 4. The generic Hilbert function is [1,2,2] with g and any two of the polynomials in u × (x − v) forming a parameterized h-basis, say…”

Section: Rulings Of a Quadricmentioning

confidence: 99%

See 2 more Smart Citations

A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

Hauenstein

Liddell

2015

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation

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Exceptional sets are the sets where the dimension of the fiber of a map is larger than the generic fiber dimension, which we assume is zero. Such situations naturally arise in kinematics, for example, when designing a mechanism that moves when the generic case is rigid. In 2008, Sommese and Wampler showed that one can use fiber products to promote such sets to become irreducible components. We propose an alternative approach using rank constraints on Macaulay matrices. Symbolic computations are used to construct the proper Macaulay matrices, while numerical computations are used to solve the rank-constraint problem. Various exceptional sets are computed, including exceptional RR dyads, lines on surfaces in C 3 , and exceptional planar pentads.

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Section: Rank-deficiency Setsmentioning

confidence: 98%

Section: Example Using Rr Dyadmentioning

confidence: 99%

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A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

Hauenstein

Liddell

2015

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation

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“…Such methods involving Gröbner bases include [10,24,31]. From the numerical algebraic geometry perspective, those in [32,2] are can be used and involve SVD decomposition and regeneration (respectively) to find critical points for the computation of witness sets. For computing the real critical points, one method is to compute all complex solutions and determine the real solutions among them.…”

Section: Introductionmentioning

confidence: 99%

Critical points via monodromy and local methods

Campo¹,

Rodriguez²

2017

Journal of Symbolic Computation

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Abstract. In many areas of applied mathematics and statistics, it is a fundamental problem to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the objective function restricted to the model.We compile ideas arising from numerical algebraic geometry to compute the critical points of an objective function. Our method consists of using numerical homotopy continuation and a monodromy action on the total critical space to compute all of the complex critical points of an objective function. To illustrate the relevance of our method, we apply it to the Euclidean distance function to compute ED-degrees and the likelihood function to compute maximum likelihood degrees.

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“…Since only the image under the projection map \pi is of interest, one can simplify this computation using [1] by not having to consider all possible slices in the auxiliary variables \Lambda \ell . This regeneration procedure produces unions of witness point sets which can be decomposed into witness points sets for the irreducible components using monodromy and a trace test, as above.…”

mentioning

confidence: 99%

Exceptional Stewart--Gough Platforms, Segre Embeddings, and the Special Euclidean Group

Hauenstein¹,

Sherman²,

Wampler³

2018

SIAM J. Appl. Algebra Geometry

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Abstract. Stewart--Gough platforms are mechanisms which consist of two rigid objects, a base and a platform, connected by six legs via spherical joints. For fixed leg lengths, a generic Stewart--Gough platform is rigid with 40 assembly configurations (over the complex numbers), while exceptional Stewart--Gough platforms have infinitely many assembly configurations and thus have self-motion. We define a family of exceptional Stewart--Gough platforms called Segre-dependent Stewart--Gough platforms which arise from a linear dependency of point-pairs under the Segre embedding and compute an irreducible decomposition of this family. We also consider Stewart--Gough platforms which move with two degrees of freedom. Since the Segre embedding arises from a representation of the special Euclidean group in three dimensions which has degree 40, we consider the special Euclidean group in other dimensions and compute spatial Stewart--Gough platforms that move in 4-dimensional space.

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Homotopies for Connected Components of Algebraic Sets with Application to Computing Critical Sets

Cited by 6 publications

References 23 publications

A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

A hybrid symbolic-numeric approach to exceptional sets of generically zero-dimensional systems

Critical points via monodromy and local methods

Exceptional Stewart--Gough Platforms, Segre Embeddings, and the Special Euclidean Group

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