Newton iteration for lexicographic Gröbner bases in two variables
É Schost, C St-Pierre - Journal of Algebra, 2024 - Elsevier
We present an m-adic Newton iteration with quadratic convergence for lexicographic
Gröbner basis of zero dimensional ideals in two variables. We rely on a structural result …
Gröbner basis of zero dimensional ideals in two variables. We rely on a structural result …
Tensor decomposition and homotopy continuation
A Bernardi, NS Daleo, JD Hauenstein… - Differential geometry and …, 2017 - Elsevier
A computationally challenging classical elimination theory problem is to compute
polynomials which vanish on the set of tensors of a given rank. By moving away from …
polynomials which vanish on the set of tensors of a given rank. By moving away from …
Smooth points on semi-algebraic sets
K Harris, JD Hauenstein, A Szanto - ACM Communications in Computer …, 2021 - dl.acm.org
Many algorithms for determining properties of semi-algebraic sets rely upon the ability to
compute smooth points [1]. We present a simple procedure based on computing the critical …
compute smooth points [1]. We present a simple procedure based on computing the critical …
A certified iterative method for isolated singular roots
In this paper we provide a new method to certify that a nearby polynomial system has a
singular isolated root and we compute its multiplicity structure. More precisely, given a …
singular isolated root and we compute its multiplicity structure. More precisely, given a …
On isolation of simple multiple zeros and clusters of zeros of polynomial systems
Given a well-constrained polynomial system $ f $ associated with a simple multiple zero $ x
$ of multiplicity $\mu $, we give a computable separation bound for isolating $ x $ from the …
$ of multiplicity $\mu $, we give a computable separation bound for isolating $ x $ from the …
Locating the closest singularity in a polynomial homotopy
J Verschelde, K Viswanathan - International Workshop on Computer …, 2022 - Springer
A polynomial homotopy is a family of polynomial systems, where the systems in the family
depend on one parameter. If for one value of the parameter we know a regular solution, then …
depend on one parameter. If for one value of the parameter we know a regular solution, then …
Improved two-step Newton's method for computing simple multiple zeros of polynomial systems
Given a polynomial system f that is associated with an isolated singular zero ξ whose
Jacobian matrix is of corank one, and an approximate zero x that is close to ξ, we propose …
Jacobian matrix is of corank one, and an approximate zero x that is close to ξ, we propose …
Change of basis for m-primary ideals in one and two variables
Following recent work by van der Hoeven and Lecerf (ISSAC 2017), we discuss the
complexity of linear mappings, called untangling and\emphtangling by those authors, that …
complexity of linear mappings, called untangling and\emphtangling by those authors, that …
Squarefree normal representation of zeros of zero-dimensional polynomial systems
J Xu, D Wang, D Lu - Journal of Symbolic Computation, 2024 - Elsevier
For any zero-dimensional polynomial ideal I and any nonzero polynomial F, this paper
shows that the union of the multi-set of zeros of the ideal sum I+< F> and that of the ideal …
shows that the union of the multi-set of zeros of the ideal sum I+< F> and that of the ideal …
Punctual Hilbert scheme and certified approximate singularities
In this paper we provide a new method to certify that a nearby polynomial system has a
singular isolated root and we compute its multiplicity structure. More precisely, given a …
singular isolated root and we compute its multiplicity structure. More precisely, given a …