18,283 research outputs found
On the algebraic variety Vr,t
AbstractThe variety Vr,t is the image under the Grassmannian map of the (t−1)-subspaces of PG(rt−1,q) of the elements of a Desarguesian spread. We investigate some properties of this variety, with particular attention to the case r=2: in this case we prove that every t+1 points of the variety are in general position and we give a new interpretation of linear sets of PG(1,qt)
Real radical initial ideals
We explore the consequences of an ideal I of real polynomials having a real
radical initial ideal, both for the geometry of the real variety of I and as an
application to sums of squares representations of polynomials. We show that if
in_w(I) is real radical for a vector w in the tropical variety, then w is in
the logarithmic set of the real variety. We also give algebraic sufficient
conditions for w to be in the logarithmic limit set of a more general
semialgebraic set. If in addition the entries of w are positive, then the
corresponding quadratic module is stable. In particular, if in_w(I) is real
radical for some positive vector w then the set of sums of squares modulo I is
stable. This provides a method for checking the conditions for stability given
by Powers and Scheiderer.Comment: 16 pages, added examples, minor revision
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
Convex Hulls of Algebraic Sets
This article describes a method to compute successive convex approximations
of the convex hull of a set of points in R^n that are the solutions to a system
of polynomial equations over the reals. The method relies on sums of squares of
polynomials and the dual theory of moment matrices. The main feature of the
technique is that all computations are done modulo the ideal generated by the
polynomials defining the set to the convexified. This work was motivated by
questions raised by Lov\'asz concerning extensions of the theta body of a graph
to arbitrary real algebraic varieties, and hence the relaxations described here
are called theta bodies. The convexification process can be seen as an
incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic
set in R^n. When the defining ideal is real radical the results become
especially nice. We provide several examples of the method and discuss
convergence issues. Finite convergence, especially after the first step of the
method, can be described explicitly for finite point sets.Comment: This article was written for the "Handbook of Semidefinite, Cone and
Polynomial Optimization: Theory, Algorithms, Software and Applications
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