Free factorization algebras and homology of configuration spaces in algebraic geometry

The configuration spaces (c-space) of mechanisms and robots can in many cases be presented as an algebraic variety. Different motion modes of mechanisms and robots are found as irreducible components of the variety. Singularities of the variety correspond usually (but not necessarily) to intersections of irreducible components/motion modes of the configuration space. A well-known method for finding the modes is the prime (and/or primary) decomposition of the constraint ideal corresponding to the mechanisms specific constraint map. However the direct computation of these decompositions is still in many cases too exhausting at least for standard computers. In this paper we present a method to speed up the decomposition significantly. If the mechanism consists or is constructed of subsystems whose c-space can be decomposed in feasible time then the whole decomposition of the c-space of the mechanism can be constructed from the decompositions of the subsystems. Here we concentrate on the 4-bar-subsystems but the approach generalizes naturally to more complicated subsystems as well. In fact the method works for all mechanisms which are constructed of subsystems whose decomposition is already known or can be computed. Further we present a way to investigate the nature of singularities which relies on the computation of the tangent cone at singular points of the c-space and the investigation of the primary decomposition of the tangent cone itself and partially its connection to intersection theory of algebraic varieties.

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Nonlinear Algebra and Applications

Numerical Algebra Control and Optimization ◽

10.3934/naco.2021045 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Paul Breiding ◽

Türkü Özlüm Çelik ◽

Timothy Duff ◽

Alexander Heaton ◽

Aida Maraj ◽

...

Keyword(s):

Algebraic Geometry ◽

Partial Differential Equations ◽

Integrable Systems ◽

Polynomial Optimization ◽

Configuration Spaces ◽

Reaction Networks ◽

Algebraic Statistics ◽

Biochemical Reaction Networks ◽

Tensor Decompositions ◽

Nonlinear Algebra

<p style='text-indent:20px;'>We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.</p>

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