Homotopies exploiting Newton polytopes for solving sparse polynomial systems
SIAM Journal on Numerical Analysis
A polyhedral method for solving sparse polynomial systems
Mathematics of Computation
Computational methods in commutative algebra and algebraic geometry
Computational methods in commutative algebra and algebraic geometry
Some lower bounds for the complexity of continuation methods
Journal of Complexity
Multivariate polynomials, duality, and structured matrices
Journal of Complexity
Solving projective complete intersection faster
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Relations between roots and coefficients, interpolation and application to system solving
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Nearest multivariate system with given root multiplicities
Journal of Symbolic Computation
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We propose an algorithm to compute simultaneously all the solutions of an algebraic system (of n equations in n variables) that define a zero-dimentional variety. This new approach generalises the univariate Weierstrass's method. We study the arithmetic complexity of this method that has a quadratic convergence in a neighbourhood of the solutions. Hereafter, we describe a method based on the iteration function of the multivariate Weierstrass's method and on the continuation method for computing the roots of polynomial systems. Finally we describe some numerical experiments of those methods.