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This paper presents algorithms for computing the multiplicity structure of a zero to a polynomial system. The zero can be exact or approximate with the system being intrinsic or empirical. As an application, the dual space theory and methodology are utilized to analyze deflation methods in solving polynomial systems, to establish tighter deflation bound,… (More)

This paper presents an algorithm and its implementation for computing the approximate GCD (greatest common divisor) of multivariate polynomials whose coefficients may be inexact. The method and the companion software appears to be the first practical package with such capabilities. The most significant features of the algorithm are its robustness and… (More)

This paper presents an algorithm and its implementation for computing the approximate GCD (greatest common divisor) of multivariate polynomials whose coefficients may be inexact. The method and the companion software appear to be the first practical package with such capabilities. The most significant features of the algorithm are its robustness and… (More)

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 a depth-deflation method for accurate computation of multiple zeros, and introduces the basic algebraic theory of the… (More)

Adapting a 1915 method of Macaulay, one can give a calculation of the local ring of an isolated zero of a polynomial system {<i>f</i><sub>1</sub>, <i>f</i><sub>2</sub>, . . . , <i>f</i><sub>t</sub>} ⊆ C[<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, . . . , <i>x</i><sub>s</sub>] using oating point arithmetic. Using an approximate reverse reduced row… (More)

- B. H. Dayton
- 1993

A symbolic-numeric method for calculating an H-basis for the ideal of a positive dimensional complex affine algebraic variety, possibly defined numerically, is given. H-bases for ideals <i>I</i> in <i>I</i>, introduced by Macaulay and later studied by Möller and Sauer, are an analog of Gröbner bases with respect to a global degree ordering:… (More)

Generic realizations of configurations of lines as reducible projective varieties in P n are constructed numerically. Examples are taken from projections of abstract seminormal configurations and the classical geometry related to Schläfli's double-six configuration. The ideal defining the homogeneous coordinate ring of a realization is calculated along with… (More)

In a recent paper with Zhonggang Zeng [2], we showed how to compute the structure of the local ring of an isolated zero from an approximate estimate of the zero, adapting a method of Macaulay (1915) by using approximate rank revealing. This can be formalized by the concept of an approximate local ring. By further relaxing the tolerance, information can be… (More)

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