Updated: July 5th, 2023 15:56

2023 journal article

A certified iterative method for isolated singular roots

*JOURNAL OF SYMBOLIC COMPUTATION*, *115*, 223–247.

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.

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.

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*.

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.

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

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.

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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