27 research outputs found
The Breadth-one -invariant Polynomial Subspace
We demonstrate the equivalence of two classes of -invariant polynomial
subspaces introduced in [8] and [9], i.e., these two classes of subspaces are
different representations of the breadth-one -invariant subspace. Moreover,
we solve the discrete approximation problem in ideal interpolation for the
breadth-one -invariant subspace. Namely, we find the points, such that the
limiting space of the evaluation functionals at these points is the functional
space induced by the given -invariant subspace, as the evaluation points all
coalesce at one point
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 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
Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals
An ideal of a local polynomial ring can be described by calculating a
standard basis with respect to a local monomial ordering. However standard
basis algorithms are not numerically stable. Instead we can describe the ideal
numerically by finding the space of dual functionals that annihilate it,
reducing the problem to one of linear algebra. There are several known
algorithms for finding the truncated dual up to any specified degree, which is
useful for describing zero-dimensional ideals. We present a stopping criterion
for positive-dimensional cases based on homogenization that guarantees all
generators of the initial monomial ideal are found. This has applications for
calculating Hilbert functions.Comment: 19 pages, 4 figure
Clustering Complex Zeros of Triangular Systems of Polynomials
This paper gives the first algorithm for finding a set of natural
-clusters of complex zeros of a triangular system of polynomials
within a given polybox in , for any given . Our
algorithm is based on a recent near-optimal algorithm of Becker et al (2016)
for clustering the complex roots of a univariate polynomial where the
coefficients are represented by number oracles.
Our algorithm is numeric, certified and based on subdivision. We implemented
it and compared it with two well-known homotopy solvers on various triangular
systems. Our solver always gives correct answers, is often faster than the
homotopy solver that often gives correct answers, and sometimes faster than the
one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update