Related papers: Real Solution Isolation with Multiplicity of Zero-…
Using the local geometrical properties of a given zero-dimensional square multivariate nonlinear system inside a box, we provide a simple but effective and new criterion for the uniqueness and the existence of a real simple zero of the…
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 present a zero decomposition theorem and an algorithm based on Wu's method, which computes a zero decomposition with multiplicity for a given zero-dimensional polynomial system. If the system satisfies some condition, the zero…
We present a new algorithm for solving the real roots of a bivariate polynomial system $\Sigma=\{f(x,y),g(x,y)\}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate…
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…
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…
As an attempt to bridge between numerical analysis and algebraic geometry, this paper formulates the multiplicity for the general nonlinear system at an isolated zero, presents an algorithm for computing the multiplicity structure, proposes…
This paper solves the open problem on the sharp bound for the number of isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an unmixed…
Multidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional…
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…
Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role…
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 $\textbf{V}(J)$…
This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. Our algorithm is based on a…
In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set's zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition…
In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input…
We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems…
We formulate a resolution of singularities algorithm for analyzing the zero sets of real-analytic functions in dimensions $\geq 3$. Rather than using the celebrated result of Hironaka, the algorithm is modeled on a more explicit and…
An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consiting of a sheaf of ideals and a positive…
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…
Triangular decomposition with different properties has been used for various types of problem solving, e.g. geometry theorem proving, real solution isolation of zero-dimensional polynomial systems, etc. In this paper, the concepts of strong…