Works (35)
Updated: July 5th, 2023 15:56
2023 journal article
A certified iterative method for isolated singular roots
JOURNAL OF SYMBOLIC COMPUTATION, 115, 223–247.
author keywords: Polynomial equations; Isolated solution; Multiple root; Certification; Newton iteration; Deflation
hal.archives-ouvertes.fr (repository)
Sources: Web Of Science, NC State University Libraries
Added: October 3, 2022
2023 journal article
Certified Hermite matrices from approximate roots
JOURNAL OF SYMBOLIC COMPUTATION, 117, 101–118.
author keywords: Symbolic-numeric computation; Polynomial systems; Approximate roots; Hermite matrices; Certification
Source: Web Of Science
Added: February 20, 2023
2023 journal article
Smooth points on semi-algebraic sets
JOURNAL OF SYMBOLIC COMPUTATION, 116, 183–212.
author keywords: Computational real algebraic geometry; Real smooth points; Real dimension; Polar varieties; Numerical algebraic geometry; Kuramoto model
TL;DR:
A simple procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected bounded component of a real atomic semi-algebraic set is presented.
(via Semantic Scholar)
Source: Web Of Science
Added: October 31, 2022
2021 article
Preface (vol 19, pg 963, 2019)
Cohen, A., Dahmen, W., Munthe-Kaas, H., Sombra, M., & Szanto, A. (2021, October 19). FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, Vol. 10.
Sources: Web Of Science, NC State University Libraries
Added: November 1, 2021
2020 conference paper
Certified Hermite Matrices from Approximate Roots - Univariate Case
Mathematical Aspects of Computer and Information Sciences, 3–9.
TL;DR:
This study describes a symbolic-numeric algorithm to construct a rational matrix, called Hermite matrix, from the approximate roots and certify that this matrix is the true Hermite matrices corresponding to the roots \(V({\mathcal I})\).
(via Semantic Scholar)
Source: ORCID
Added: November 6, 2020
2020 conference paper
Certified Hermite Matrices from Approximate Roots - Univariate Case
Mathematical Aspects of Computer and Information Sciences, 3–9.
Source: ORCID
Added: November 13, 2020
2020 conference paper
Punctual Hilbert scheme and certified approximate singularities
Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation.
TL;DR:
A new method to certify that a nearby polynomial system has a singular isolated root and it is found that some of its strata can be rationally parametrized and exploit these parametrizations in the certification.
(via Semantic Scholar)
Source: ORCID
Added: November 6, 2020
2020 journal article
Smooth Points on Semi-algebraic Sets
ACM COMMUNICATIONS IN COMPUTER ALGEBRA, 54(3), 105–108.
Source: Web Of Science
Added: April 12, 2021
2020 article
Smooth Points on Semi-algebraic Sets
(2020, February 11).
Source: ORCID
Added: November 6, 2020
2020 journal article
Subresultants of (x - alpha)(m) and (x - beta)(n), Jacobipolynomials and complexity
JOURNAL OF SYMBOLIC COMPUTATION, 101, 330–351.
author keywords: Subresultants; Algorithms; Complexity; Jacobi polynomials
TL;DR:
It is shown that the coefficients of the subresultants of ( x − α ) m and (x − β ) n with respect to the monomial basis can be computed in linear arithmetic complexity, which is faster than for arbitrary polynomials.
(via Semantic Scholar)
Sources: Web Of Science, ORCID, NC State University Libraries
Added: July 6, 2020
2019 journal article
Closed formula for univariate subresultants in multiple roots
LINEAR ALGEBRA AND ITS APPLICATIONS, 565, 123–155.
author keywords: Subresultants; Exchange lemma; Formulas in roots; Schur functions
Sources: Web Of Science, ORCID, NC State University Libraries
Added: February 25, 2019
2018 journal article
Certifying solutions to overdetermined and singular polynomial systems over Q
JOURNAL OF SYMBOLIC COMPUTATION, 84, 147–171.
author keywords: Overdetermined; Polynomial system; Singular solutions; Certification; Rational univariate representation; Isosingular deflation
TL;DR:
This papercertifies that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients, and uses hybrid symbolic-numeric methods to compute the exact "rational univariate representation" of a component of the input system from approximate roots.
(via Semantic Scholar)
Sources: Web Of Science, ORCID, NC State University Libraries
Added: August 6, 2018
2018 article
Irredundant Triangular Decomposition
ISSAC'18: PROCEEDINGS OF THE 2018 ACM INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION, pp. 311–318.
TL;DR:
This paper gives an algorithm to compute an irredundant triangular decomposition of an arbitrary algebraic set W defined by a set of polynomials in C[x1, x2, ..., xn]; this algorithm is randomized, and the probability of success is analyzed.
(via Semantic Scholar)
Sources: Web Of Science, ORCID, NC State University Libraries
Added: February 25, 2019
2017 article
On deflation and multiplicity structure
JOURNAL OF SYMBOLIC COMPUTATION, Vol. 83, pp. 228–253.
author keywords: Deflation; Multiplicity structure; Newton's method; Inverse system; Multiplication matrix
TL;DR:
Two new constructions related to singular solutions of polynomial systems are presented, one of which gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root and the other allows one to treat all conjugate roots simultaneously.
(via Semantic Scholar)
Sources: Web Of Science, ORCID, NC State University Libraries
Added: August 6, 2018
2017 journal article
Subresultants in multiple roots: An extremal case
Linear Algebra and Its Applications, 529, 185–198.
author keywords: Subresultants; Hankel matrices; Ostrowski's determinant; Pfaff-Saalschiltz identity
Sources: Crossref, ORCID, NC State University Libraries
Added: August 28, 2020
2017 journal article
Symmetric interpolation, Exchange Lemma and Sylvester sums
COMMUNICATIONS IN ALGEBRA, 45(8), 3231–3250.
author keywords: Subresultants; Sylvester double sums; symmetric Lagrange interpolation
Sources: Web Of Science, ORCID, NC State University Libraries
Added: August 6, 2018
2016 article
Special issue on the conference ISSAC 2014: Symbolic computation and computer algebra Foreword
JOURNAL OF SYMBOLIC COMPUTATION, Vol. 75, pp. 1–3.
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2015 conference paper
Certifying Isolated Singular Points and their Multiplicity Structure
Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '15.
author keywords: isolated point; root deflation; dual space; multiplicity structure; inverse system; local algebra; multiplication operator
TL;DR:
Two new constructions related to singular solutions of polynomial systems are presented, one of which gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root and the other presents a system of equations in the original variables plus a relatively small number of new variables.
(via Semantic Scholar)
Source: ORCID
Added: November 6, 2020
2015 journal article
Overdetermined Weierstrass iteration and the nearest consistent system
THEORETICAL COMPUTER SCIENCE, 562, 346–364.
author keywords: Approximate GCD; Overdetermined systems; Nearest consistent system; Weierstrass-Durand-Kerner method
TL;DR:
A generalization of the Weierstrass iteration for overdetermined systems of equations and it is proved that the proposed method is the Gauss-Newton iteration to find the nearest system which has at least k common roots and which is obtained via a perturbation of prescribed structure.
(via Semantic Scholar)
Sources: Web Of Science, ORCID, NC State University Libraries
Added: August 6, 2018
2015 journal article
Subresultants, Sylvester sums and the rational interpolation problem
JOURNAL OF SYMBOLIC COMPUTATION, 68, 72–83.
author keywords: Rational interpolation; Cauchy interpolation; Osculatory interpolation; Rational Hermite interpolation; Subresultants; Sylvester sums
TL;DR:
A solution for the classical univariate rational interpolation problem by means of (univariate) subresultants is presented, giving explicit formulas for the solution in terms of symmetric functions of the input data, generalizing the well-known formulas for Lagrange interpolation.
(via Semantic Scholar)
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2014 chapter
A Note on Global Newton Iteration Over Archimedean and Non-Archimedean Fields
In Computer Algebra in Scientific Computing (Vol. 8660, pp. 202–217).
Sources: Crossref, NC State University Libraries
Added: August 6, 2018
2014 conference paper
A Note on Global Newton Iteration Over Archimedean and Non-Archimedean Fields
In V. P. Gerdt, W. Koepf, W. M. Seiler, & E. V. Vorozhtsov (Eds.), Computer Algebra in Scientific Computing (pp. 202–217). Cham: Springer International Publishing.
Ed(s): V. Gerdt, W. Koepf, W. Seiler & E. Vorozhtsov
Event: at Cham
Source: ORCID
Added: November 6, 2020
2013 journal article
Subresultants in multiple roots
LINEAR ALGEBRA AND ITS APPLICATIONS, 438(5), 1969–1989.
author keywords: subresultants; Sylvester sums; root multiplicity; Hermite interpolation
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2012 journal article
On the computation of matrices of traces and radicals of ideals
Journal of Symbolic Computation, 47(1), 102–122.
author keywords: Matrix of traces; Radical of an ideal
TL;DR:
Borders on the degrees needed for the Macaulay matrix are proved in the case when I has finitely many projective roots in P"K^m" and previous results which work only for the case where A is Gorenstein to the non-Gorenstein case are extended.
(via Semantic Scholar)
Sources: Crossref, NC State University Libraries
Added: June 15, 2021
2012 journal article
Sylvester's double sums: An inductive proof of the general case
JOURNAL OF SYMBOLIC COMPUTATION, 47(8), 942–953.
author keywords: Sylvester's double sums; Subresultants
TL;DR:
It is shown how induction also allows to obtain the full description of Sylvester?s double-sums.
(via Semantic Scholar)
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2011 chapter
Symbolic-Numeric Solution of Ill-Conditioned Polynomial Systems (Survey Talk Overview) (Invited Talk)
In V. P. Gerdt, W. Koepf, E. W. Mayr, & E. V. Vorozhtsov (Eds.), Computer Algebra in Scientific Computing (CASC 2011) (pp. 345–347).
Ed(s): V. Gerdt, W. Koepf, E. Mayr & E. Vorozhtsov
TL;DR:
A family of iterative techniques which, for a given inexact system of polynomials and given root structure, computes the nearest system which has roots with the given structure are described.
(via Semantic Scholar)
Sources: Crossref, NC State University Libraries
Added: June 15, 2021
2010 journal article
Multivariate subresultants using Jouanolou matrices
JOURNAL OF PURE AND APPLIED ALGEBRA, 214(8), 1347–1369.
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2009 journal article
A bound for orders in differential Nullstellensatz
JOURNAL OF ALGEBRA, 322(11), 3852–3877.
author keywords: Differential algebra; Characteristic sets; Radical differential ideals; Differential Nullstellensatz
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2009 journal article
Nearest multivariate system with given root multiplicities
JOURNAL OF SYMBOLIC COMPUTATION, 44(6), 606–625.
author keywords: Approximate polynomial systems; Multiple roots; Multiplicity structure; Weierstrass iteration
TL;DR:
A symbolic-numeric technique to find the closest multivariate polynomial system to a given one which has roots with prescribed multiplicity structure and a simplified version of the iteration function analogously to the classical Weierstrass iteration, which allows a component-wise expression, and thus reduces the computational cost of each iteration.
(via Semantic Scholar)
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2009 journal article
Sylvester’s double sums: The general case
Journal of Symbolic Computation, 44(9), 1164–1175.
author keywords: Subresultants; Double sums; Vandermonde determinants
TL;DR:
The technique developed to answer the question of what are the other members of the Sylvester family turns out to be general enough to characterize allMembers of the family, providing a uniform method.
(via Semantic Scholar)
Sources: Crossref, NC State University Libraries, Web Of Science, ORCID
Added: August 6, 2018
2008 journal article
Solving over-determined systems by the subresultant method (with an appendix by Marc Chardin)
JOURNAL OF SYMBOLIC COMPUTATION, 43(1), 46–74.
author keywords: multivariate subresultant; over-determined polynomial system; solution of polynomial system
TL;DR:
It is proved that certain previously known matrix constructions, in particular, Macaulay's, Chardin's and Jouanolou's resultant and sub resultant matrices possess the subresultant properties.
(via Semantic Scholar)
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2007 journal article
An elementary proof of Sylvester's double sums for subresultants
JOURNAL OF SYMBOLIC COMPUTATION, 42(3), 290–297.
author keywords: subresultants; double-sum formula; Vandermonde determinant
TL;DR:
An elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants to express the polynomial subresultants in terms of the roots of the input polynomials is provided.
(via Semantic Scholar)
Sources: Web Of Science, NC State University Libraries, ORCID
Added: August 6, 2018
2007 journal article
Approximate Radical for Clusters: A Global Approach Using Gaussian Elimination or SVD
Mathematics in Computer Science, 1(2), 393–425.
author keywords: Radical ideal; clusters; matrix of traces; symbolic-numeric computation
TL;DR:
A method based on Dickson’s lemma to compute the “approximate radical” of the matrix of traces, attached to a zero dimensional ideal which has zero clusters, is presented.
(via Semantic Scholar)
Sources: Crossref, NC State University Libraries
Added: June 15, 2021
2006 journal article
Multivariate subresultants in roots
JOURNAL OF ALGEBRA, 302(1), 16–36.
author keywords: subresultants; Poisson product formula; Vandermonde determinants
Sources: Web Of Science, NC State University Libraries
Added: August 6, 2018
2006 journal article
Re: “Multivariate subresultants in roots” [J. Algebra 302 (2006) 16–36]
Journal of Algebra, 303(2), 449.
Sources: Crossref, NC State University Libraries
Added: August 28, 2020
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