165 research outputs found
Solving parametric systems of polynomial equations over the reals through Hermite matrices
We design a new algorithm for solving parametric systems having finitely many
complex solutions for generic values of the parameters. More precisely, let with and
, be the algebraic set
defined by and be the projection . Under the
assumptions that admits finitely many complex roots for generic values of
and that the ideal generated by is radical, we solve the following
problem. On input , we compute semi-algebraic formulas defining
semi-algebraic subsets of the -space such that
is dense in and the number of real points in
is invariant when varies over each .
This algorithm exploits properties of some well chosen monomial bases in the
algebra where is the ideal generated by in
and the specialization property of the so-called Hermite
matrices. This allows us to obtain compact representations of the sets by
means of semi-algebraic formulas encoding the signature of a symmetric matrix.
When satisfies extra genericity assumptions, we derive complexity bounds on
the number of arithmetic operations in and the degree of the
output polynomials. Let be the maximal degree of the 's and , we prove that, on a generic , one can compute
those semi-algebraic formulas with operations in and that the polynomials involved
have degree bounded by .
We report on practical experiments which illustrate the efficiency of our
algorithm on generic systems and systems from applications. It allows us to
solve problems which are out of reach of the state-of-the-art
CHAMP: A Cherednik Algebra Magma Package
We present a computer algebra package based on Magma for performing
computations in rational Cherednik algebras at arbitrary parameters and in
Verma modules for restricted rational Cherednik algebras. Part of this package
is a new general Las Vegas algorithm for computing the head and the
constituents of a module with simple head in characteristic zero which we
develop here theoretically. This algorithm is very successful when applied to
Verma modules for restricted rational Cherednik algebras and it allows us to
answer several questions posed by Gordon in some specific cases. We could
determine the decomposition matrices of the Verma modules, the graded G-module
structure of the simple modules, and the Calogero-Moser families of the generic
restricted rational Cherednik algebra for around half of the exceptional
complex reflection groups. In this way we could also confirm Martino's
conjecture for several exceptional complex reflection groups.Comment: Final version to appear in LMS J. Comput. Math. 41 pages, 3 ancillary
files. CHAMP is available at http://thielul.github.io/CHAMP/. All results are
listed explicitly in the ancillary PDF document (currently 935 pages). Please
check the website for further update
Segre Class Computation and Practical Applications
Let be closed (possibly singular) subschemes of a smooth
projective toric variety . We show how to compute the Segre class
as a class in the Chow group of . Building on this, we give effective
methods to compute intersection products in projective varieties, to determine
algebraic multiplicity without working in local rings, and to test pairwise
containment of subvarieties of . Our methods may be implemented without
using Groebner bases; in particular any algorithm to compute the number of
solutions of a zero-dimensional polynomial system may be used
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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