Gröbner bases and primary decomposition of polynomial ideals
Journal of Symbolic Computation
Solving zero-dimensional algebraic systems
Journal of Symbolic Computation
A generalized Euclidean algorithm for computing triangular representations of algebraic varieties
Journal of Symbolic Computation
Complexity and real computation
Complexity and real computation
Singular systems of polynomials
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
On the theories of triangular sets
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Triangular sets for solving polynomial systems: a comparative implementation of four methods
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Numerical homotopies to compute generic points on positive dimensional algebraic sets
Journal of Complexity
Using Galois ideals for computing relative resolvents
Journal of Symbolic Computation - Algorithmic methods in Galois Theory
The nearest polynomial with a given zero, and similar problems
ACM SIGSAM Bulletin
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
SIAM Journal on Numerical Analysis
Complexity results for triangular sets
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Numerical Polynomial Algebra
Sharp estimates for triangular sets
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Lifting techniques for triangular decompositions
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Computing the multiplicity structure in solving polynomial systems
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Approximate radical of ideals with clusters of roots
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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Triangular decompositions for systems of polynomial equations with n variables, with exact coefficients, are well developed theoretically and in terms of implemented algorithms in computer algebra systems. However there is much less research concerning triangular decompositions for systems with approximate coefficients. In this paper we discuss the zero-dimensional case of systems having finitely many roots. Our methods depend on having approximations for all the roots, and these are provided by the homotopy continuation methods of Sommese, Verschelde and Wampler. We introduce approximate equiprojectable decompositions for such systems, which represent a generalization of the recently developed analogous concept for exact systems. We demonstrate experimentally the favorable computational features of this new approach, and give a statistical analysis of its error.