160 research outputs found
On the complexity of computing Gr\"obner bases for weighted homogeneous systems
Solving polynomial systems arising from applications is frequently made
easier by the structure of the systems. Weighted homogeneity (or
quasi-homogeneity) is one example of such a structure: given a system of
weights , -homogeneous polynomials are polynomials
which are homogeneous w.r.t the weighted degree
. Gr\"obner bases for weighted homogeneous systems can be
computed by adapting existing algorithms for homogeneous systems to the
weighted homogeneous case. We show that in this case, the complexity estimate
for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be
divided by a factor . For zero-dimensional
systems, the complexity of Algorithm~\FGLM (where is the
number of solutions of the system) can be divided by the same factor
. Under genericity assumptions, for
zero-dimensional weighted homogeneous systems of -degree
, these complexity estimates are polynomial in the
weighted B\'ezout bound .
Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by
the weighted Macaulay bound , and this bound is
sharp if we can order the weights so that . For overdetermined
semi-regular systems, estimates from the homogeneous case can be adapted to the
weighted case. We provide some experimental results based on systems arising
from a cryptography problem and from polynomial inversion problems. They show
that taking advantage of the weighted homogeneous structure yields substantial
speed-ups, and allows us to solve systems which were otherwise out of reach
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
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
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