Related papers: Parallel computation of real solving bivariate pol…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm BISOLVE. BISOLVE is a classical elimination method which first projects the solutions of a system onto the $x$-…
Existing algorithms for isolating real solutions of zero-dimensional polynomial systems do not compute the multiplicities of the solutions. In this paper, we define in a natural way the multiplicity of solutions of zero-dimensional…
We propose a symbolic-numeric algorithm to count the number of solutions of a polynomial system within a local region. More specifically, given a zero-dimensional system $f_1=\cdots=f_n=0$, with $f_i\in\mathbb{C}[x_1,\ldots,x_n]$, and a…
We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…
We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows a framework developed in previous work and can operate in…
This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present…
In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using…
Let $f$ be a polynomial system consisting of $n$ polynomials $f_1,\cdots, f_n$ in $n$ variables $x_1,\cdots, x_n$, with coefficients in $\mathbb{Q}$ and let $\langle f\rangle$ be the ideal generated by $f$. Such a polynomial system, which…
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…
We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the…
We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…
Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach…
In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate…
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…
This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final…
In this paper we apply for the first time a new method for multivariate equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for complex root determination to the {\em real} case. Our main result concerns the problem…
We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…