53 research outputs found
On isolation of singular zeros of multivariate analytic systems
We give a separation bound for an isolated multiple root of a square
multivariate analytic system satisfying that an operator deduced by adding
and a projection of in a direction of the kernel of
is invertible. We prove that the deflation process applied on and this kind
of roots terminates after only one iteration. When is only given
approximately, we give a numerical criterion for isolating a cluster of zeros
of near . We also propose a lower bound of the number of roots in the
cluster.Comment: 17 page
Certifying isolated singular points and their multiplicity structure
This paper presents two new constructions related to singular solutions of
polynomial systems. The first is a new deflation method for an isolated
singular root. This construc-tion uses a single linear differential form
defined from the Jacobian matrix of the input, and defines the deflated system
by applying this differential form to the original system. The advantages of
this new deflation is that it does not introduce new variables and the increase
in the number of equations is linear instead of the quadratic increase of
previous methods. The second construction gives the coefficients of the
so-called inverse system or dual basis, which defines the multiplicity
structure at the singular root. We present a system of equations in the
original variables plus a relatively small number of new vari-ables. We show
that the roots of this new system include the original singular root but now
with multiplicity one, and the new variables uniquely determine the
multiplicity structure. Both constructions are "exact", meaning that they
permit one to treat all conjugate roots simultaneously and can be used in
certification procedures for singular roots and their multiplicity structure
with respect to an exact rational polynomial system
On deflation and multiplicity structure
This paper presents two new constructions related to singular solutions of
polynomial systems. The first is a new deflation method for an isolated
singular root. This construction uses a single linear differential form defined
from the Jacobian matrix of the input, and defines the deflated system by
applying this differential form to the original system. The advantages of this
new deflation is that it does not introduce new variables and the increase in
the number of equations is linear in each iteration instead of the quadratic
increase of previous methods. The second construction gives the coefficients of
the so-called inverse system or dual basis, which defines the multiplicity
structure at the singular root. We present a system of equations in the
original variables plus a relatively small number of new variables that
completely deflates the root in one step. We show that the isolated simple
solutions of this new system correspond to roots of the original system with
given multiplicity structure up to a given order. Both constructions are
"exact" in that they permit one to treat all conjugate roots simultaneously and
can be used in certification procedures for singular roots and their
multiplicity structure with respect to an exact rational polynomial system.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0508
Punctual Hilbert Schemes and Certified Approximate Singularities
In this paper we provide a new method to certify that a nearby polynomial
system has a singular isolated root with a prescribed multiplicity structure.
More precisely, given a polynomial system f , we present a Newton iteration on an extended deflated system
that locally converges, under regularity conditions, to a small deformation of
such that this deformed system has an exact singular root. The iteration
simultaneously converges to the coordinates of the singular root and the
coefficients of the so called inverse system that describes the multiplicity
structure at the root. We use -theory test to certify the quadratic
convergence, and togive bounds on the size of the deformation and on the
approximation error. The approach relies on an analysis of the punctual Hilbert
scheme, for which we provide a new description. We show in particular that some
of its strata can be rationally parametrized and exploit these parametrizations
in the certification. We show in numerical experimentation how the approximate
inverse system can be computed as a starting point of the Newton iterations and
the fast numerical convergence to the singular root with its multiplicity
structure, certified by our criteria.Comment: International Symposium on Symbolic and Algebraic Computation, Jul
2020, Kalamata, Franc
Numeric certified algorithm for the topology of resultant and discriminant curves
Let be a real plane algebraic curve defined by the resultant of
two polynomials (resp. by the discriminant of a polynomial). Geometrically such
a curve is the projection of the intersection of the surfaces
(resp. ), and generically its singularities are nodes (resp. nodes and
ordinary cusps). State-of-the-art numerical algorithms compute the topology of
smooth curves but usually fail to certify the topology of singular ones. The
main challenge is to find practical numerical criteria that guarantee the
existence and the uniqueness of a singularity inside a given box , while
ensuring that does not contain any closed loop of . We solve
this problem by first providing a square deflation system, based on
subresultants, that can be used to certify numerically whether contains a
unique singularity or not. Then we introduce a numeric adaptive separation
criterion based on interval arithmetic to ensure that the topology of in is homeomorphic to the local topology at . Our algorithms are
implemented and experiments show their efficiency compared to state-of-the-art
symbolic or homotopic methods
Verified Error Bounds for Isolated Singular Solutions of Polynomial Systems: Case of Breadth One
In this paper we describe how to improve the performance of the
symbolic-numeric method in (Li and Zhi,2009, 2011) for computing the
multiplicity structure and refining approximate isolated singular solutions in
the breadth one case. By introducing a parameterized and deflated system with
smoothing parameters, we generalize the algorithm in (Rump and Graillat, 2009)
to compute verified error bounds such that a slightly perturbed polynomial
system is guaranteed to have a breadth-one multiple root within the computed
bounds.Comment: 20 page
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