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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)

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)

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)

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)

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)

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