Gröbner bases and primary decomposition of polynomial ideals
Journal of Symbolic Computation
Strategy-accurate parallel Buchberger algorithms
Journal of Symbolic Computation - Special issue on parallel symbolic computation
Localization and primary decomposition of polynomial ideals
Journal of Symbolic Computation
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
SIAM Journal on Numerical Analysis
Yet Another Ideal Decomposition Algorithm
AAECC-12 Proceedings of the 12th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Computing the multiplicity structure in solving polynomial systems
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Newton's method with deflation for isolated singularities of polynomial systems
Theoretical Computer Science
Evaluation techniques for zero-dimensional primary decomposition
Journal of Symbolic Computation
Chern numbers of smooth varieties via homotopy continuation and intersection theory
Journal of Symbolic Computation
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Consider an ideal I ⊂ R = C[x1,...,xn] defining a complex affine variety X ⊂ Cn. We describe the components associated to I by means of numerical primary decomposition (NPD). The method is based on the construction of deflation ideal I(d) that defines the deflated variety X(d) in a complex space of higher dimension. For every embedded component there exists d and an isolated component Y(d) of I(d) projecting onto Y. In turn, Y(d) can be discovered by existing methods for prime decomposition, in particular, the numerical irreducible decomposition, applied to X(d). The concept of NPD gives a full description of the scheme Spec(R/I) by representing each component with a witness set. We propose an algorithm to produce a collection of witness sets that contains a NPD and that can be used to solve the ideal membership problem for I.