Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
A Gröbner free alternative for polynomial system solving
Journal of Complexity
Kronecker's and Newton's approaches to solving: a first comparison
Journal of Complexity
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
SIAM Journal on Numerical Analysis
Double Description Method Revisited
Selected papers from the 8th Franco-Japanese and 4th Franco-Chinese Conference on Combinatorics and Computer Science
Journal of Symbolic Computation
Counting solutions to binomial complete intersections
Journal of Complexity
Multi-variate polynomials and Newton-Puiseux expansions
SNSC'01 Proceedings of the 2nd international conference on Symbolic and numerical scientific computation
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In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic sets. The series expansions originate at the intersection of the algebraic set with as many coordinate planes as the dimension of the algebraic set. Our approach starts with a polyhedral method to compute cones of normal vectors to the Newton polytopes of the given polynomial system that defines the algebraic set. If as many vectors in the cone as the dimension of the algebraic set define an initial form system that has isolated solutions, then those vectors are potential tropisms for the initial term of the Puiseux series expansion. Our preliminary methods produce exact representations for solution sets of the cyclic n-roots problem, for n = m2, corresponding to a result of Backelin.