2 research outputs found
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