Numerically intersecting algebraic varieties via witness sets

Journal of Symbolic Computation

2017

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Section: Numerical Algebraic Geometrymentioning

confidence: 99%

A hybrid symbolic-numerical approach to the center-focus problem

Mahdi

Pessoa

Journal of Symbolic Computation

2017

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“…When extrinsic wins, it is because the number of elements in the representation of a basis for the kernel is large, which raises the expense of working intrinsically. Intrinsic implementations of homotopies are discussed in [43,20,21] and specifics on assessing the trade-off between extrinsic and intrinsic formulations can be found in [23]. The key to overcoming these shortcomings and enabling the computation of a numerical irreducible decomposition for Z Vpf q is the theory of isosingular sets [22].…”

Section: Extrinsic and Intrinsic Homotopymentioning

confidence: 99%

“…The key step of constructing system f w depends on isosingular deflation, for which we provide only the key concepts here. We refer the interested reader to [22] for a general overview with [23] providing details related to diagonal intersection. For a polynomial system G and a point z VpGq C N , Iso G pzq is an irreducible algebraic subset of VpGq containing z.…”

Section: Completing the Decompositionmentioning

confidence: 99%

Unification and extension of intersection algorithms in numerical algebraic geometry

Applied Mathematics and Computation

Wampler

2017

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The solution set of a system of polynomial equations, called an algebraic set, can be decomposed into finitely many irreducible components. In numerical algebraic geometry, irreducible algebraic sets are represented by witness sets, whereas general algebraic sets allow a numerical irreducible decomposition comprising a collection of witness sets, one for each irreducible component. We denote the solution set of any system of polynomials f : C N Ñ C n as Vpf q C N. Given a witness set for some algebraic set Z C N and a system of polynomials f : C N Ñ C n , the algorithms of this article compute a numerical irreducible decomposition of the set Z Vpf q. While extending the types of intersection problems that can be solved via numerical algebraic geometry, this approach is also a unification of two existing algorithms: the diagonal intersection algorithm and the homotopy membership test. The new approach includes as a special case the "extension problem" where one wishes to intersect an irreducible component A of Vpgpxqq with Vpf px, yqq, where f introduces new variables, y. For example, this problem arises in computing the singularities of A when the singularity conditions are expressed in terms of new variables associated to the tangent space of A. Several examples are included to demonstrate the effectiveness of our approach applied in a variety of scenarios.

Isosingular Sets and Deflation

Wampler

2013

Found Comput Math

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This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations that share a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which are similar to the sequences arising in Thom-Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic sets that can be investigated with methods from numerical algebraic geometry.