421 research outputs found
Semi-inverted linear spaces and an analogue of the broken circuit complex
The image of a linear space under inversion of some coordinates is an affine
variety whose structure is governed by an underlying hyperplane arrangement. In
this paper, we generalize work by Proudfoot and Speyer to show that circuit
polynomials form a universal Groebner basis for the ideal of polynomials
vanishing on this variety. The proof relies on degenerations to the
Stanley-Reisner ideal of a simplicial complex determined by the underlying
matroid. If the linear space is real, then the semi-inverted linear space is
also an example of a hyperbolic variety, meaning that all of its intersection
points with a large family of linear spaces are real.Comment: 16 pages, 1 figure, minor revisions and added connections to the
external activity complex of a matroi
On Computing the Elimination Ideal Using Resultants with Applications to Gr\"obner Bases
Resultants and Gr\"obner bases are crucial tools in studying polynomial
elimination theory. We investigate relations between the variety of the
resultant of two polynomials and the variety of the ideal they generate. Then
we focus on the bivariate case, in which the elimination ideal is principal. We
study - by means of elementary tools - the difference between the multiplicity
of the factors of the generator of the elimination ideal and the multiplicity
of the factors of the resultant.Comment: 7 page
Time Delay Interferometry for LISA with one arm dysfunctional
In order to attain the requisite sensitivity for LISA - a joint space mission
of the ESA and NASA- the laser frequency noise must be suppressed below the
secondary noises such as the optical path noise, acceleration noise etc. By
combining six appropriately time-delayed data streams containing fractional
Doppler shifts - a technique called time delay interferometry (TDI) - the laser
frequency noise may be adequately suppressed. We consider the general model of
LISA where the armlengths vary with time, so that second generation TDI are
relevant. However, we must envisage the possibility, that not all the optical
links of LISA will be operating at all times, and therefore, we here consider
the case of LISA operating with two arms only. As shown earlier in the
literature, obtaining even approximate solutions of TDI to the general problem
is very difficult. Since here only four optical links are relevant, the
algebraic problem simplifies considerably. We are then able to exhibit a large
number of solutions (from mathematical point of view an infinite number) and
further present an algorithm to generate these solutions
A polyhedral approach to computing border bases
Border bases can be considered to be the natural extension of Gr\"obner bases
that have several advantages. Unfortunately, to date the classical border basis
algorithm relies on (degree-compatible) term orderings and implicitly on
reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow
for calculating border bases for arbitrary degree-compatible order ideals,
which is \emph{independent} from term orderings. Moreover, the algorithm also
supports calculating degree-compatible order ideals with \emph{preference} on
contained elements, even though finding a preferred order ideal is NP-hard.
Effectively we retain degree-compatibility only to successively extend our
computation degree-by-degree. The adaptation is based on our polyhedral
characterization: order ideals that support a border basis correspond
one-to-one to integral points of the order ideal polytope. This establishes a
crucial connection between the ideal and the combinatorial structure of the
associated factor spaces
Determinantal sets, singularities and application to optimal control in medical imagery
Control theory has recently been involved in the field of nuclear magnetic
resonance imagery. The goal is to control the magnetic field optimally in order
to improve the contrast between two biological matters on the pictures.
Geometric optimal control leads us here to analyze mero-morphic vector fields
depending upon physical parameters , and having their singularities defined by
a deter-minantal variety. The involved matrix has polynomial entries with
respect to both the state variables and the parameters. Taking into account the
physical constraints of the problem, one needs to classify, with respect to the
parameters, the number of real singularities lying in some prescribed
semi-algebraic set. We develop a dedicated algorithm for real root
classification of the singularities of the rank defects of a polynomial matrix,
cut with a given semi-algebraic set. The algorithm works under some genericity
assumptions which are easy to check. These assumptions are not so restrictive
and are satisfied in the aforementioned application. As more general strategies
for real root classification do, our algorithm needs to compute the critical
loci of some maps, intersections with the boundary of the semi-algebraic
domain, etc. In order to compute these objects, the determinantal structure is
exploited through a stratifi-cation by the rank of the polynomial matrix. This
speeds up the computations by a factor 100. Furthermore, our implementation is
able to solve the application in medical imagery, which was out of reach of
more general algorithms for real root classification. For instance,
computational results show that the contrast problem where one of the matters
is water is partitioned into three distinct classes
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