Related papers: Zero Decomposition with Multiplicity of Zero-Dimen…
A zero-dimensional polynomial ideal may have a lot of complex zeros. But sometimes, only some of them are needed. In this paper, for a zero-dimensional ideal $I$, we study its complex zeros that locate in another variety $\textbf{V}(J)$…
We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.
Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the…
Efficient characteristic set methods for computing solutions of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of solutions of a proper…
A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods…
Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
In this paper, we present a generic parametrization of generically zero-dimensional parametric polynomial systems. More specifically, we study the specialization properties of the Rational Univariate Representation and derive bounds on the…
In this paper some algorithms will be presented which can be used for the calculation of zeros of polynomials and eigenvalues of polynomial matrices with a multiplicity larger than one. The numerical values calculated with MATLAB are used…
In this paper the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style are studied when the input polynomial set to decompose has a chordal associated graph. In particular, we prove that the…
In this thesis, a new class of algorithms based on Sums of Squares Programming is developed. These allow to reduce a degree-$d$ homogeneous polynomial $T = \sum_{i = 1}^m \langle a_i, X \rangle^d $ to a quadratic form being close to a…
In a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605, 2008) we analyzed a numerical algorithm for computing the number of real zeros of a…
A new concept, decomposition-unstable (DU) variety of a parametric polynomial system, is introduced in this paper and the stabilities of several triangular decomposition methods, such as characteristic set decomposition, relatively…
Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szanto. In this paper, we give the first complete bounds for the…
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…
A study of zero-dimensional theories, based on exact results, is presented. First, relying on a simple diagrammatic representation of the theory, equations involving the generating function of all connected Green's functions are…
In this paper, we first prove that when the associated graph of a polynomial set is chordal, a particular triangular set computed by a general algorithm in top-down style for computing the triangular decomposition of this polynomial set has…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
We give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components.
A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the…