468 research outputs found
Reducing the size and number of linear programs in a dynamic Gr\"obner basis algorithm
The dynamic algorithm to compute a Gr\"obner basis is nearly twenty years
old, yet it seems to have arrived stillborn; aside from two initial
publications, there have been no published followups. One reason for this may
be that, at first glance, the added overhead seems to outweigh the benefit; the
algorithm must solve many linear programs with many linear constraints. This
paper describes two methods of reducing the cost substantially, answering the
problem effectively.Comment: 11 figures, of which half are algorithms; submitted to journal for
refereeing, December 201
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
A Purely Functional Computer Algebra System Embedded in Haskell
We demonstrate how methods in Functional Programming can be used to implement
a computer algebra system. As a proof-of-concept, we present the
computational-algebra package. It is a computer algebra system implemented as
an embedded domain-specific language in Haskell, a purely functional
programming language. Utilising methods in functional programming and prominent
features of Haskell, this library achieves safety, composability, and
correctness at the same time. To demonstrate the advantages of our approach, we
have implemented advanced Gr\"{o}bner basis algorithms, such as Faug\`{e}re's
and , in a composable way.Comment: 16 pages, Accepted to CASC 201
Upgraded methods for the effective computation of marked schemes on a strongly stable ideal
Let be a monomial strongly stable ideal. The
collection \Mf(J) of the homogeneous polynomial ideals , such that the
monomials outside form a -vector basis of , is called a {\em
-marked family}. It can be endowed with a structure of affine scheme, called
a {\em -marked scheme}. For special ideals , -marked schemes provide
an open cover of the Hilbert scheme \hilbp, where is the Hilbert
polynomial of . Those ideals more suitable to this aim are the
-truncation ideals generated by the monomials of
degree in a saturated strongly stable monomial ideal .
Exploiting a characterization of the ideals in \Mf(\underline{J}_{\geq m}) in
terms of a Buchberger-like criterion, we compute the equations defining the
-marked scheme by a new reduction relation, called {\em
superminimal reduction}, and obtain an embedding of \Mf(\underline{J}_{\geq
m}) in an affine space of low dimension. In this setting, explicit
computations are achievable in many non-trivial cases. Moreover, for every ,
we give a closed embedding \phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow
\Mf(\underline{J}_{\geq m+1}), characterize those that are
isomorphisms in terms of the monomial basis of , especially we
characterize the minimum integer such that is an isomorphism for
every .Comment: 28 pages; this paper contains and extends the second part of the
paper posed at arXiv:0909.2184v2[math.AG]; sections are now reorganized and
the general presentation of the paper is improved. Final version accepted for
publicatio
Numerical Algorithms for Dual Bases of Positive-Dimensional Ideals
An ideal of a local polynomial ring can be described by calculating a
standard basis with respect to a local monomial ordering. However standard
basis algorithms are not numerically stable. Instead we can describe the ideal
numerically by finding the space of dual functionals that annihilate it,
reducing the problem to one of linear algebra. There are several known
algorithms for finding the truncated dual up to any specified degree, which is
useful for describing zero-dimensional ideals. We present a stopping criterion
for positive-dimensional cases based on homogenization that guarantees all
generators of the initial monomial ideal are found. This has applications for
calculating Hilbert functions.Comment: 19 pages, 4 figure
Dual-to-kernel learning with ideals
In this paper, we propose a theory which unifies kernel learning and symbolic
algebraic methods. We show that both worlds are inherently dual to each other,
and we use this duality to combine the structure-awareness of algebraic methods
with the efficiency and generality of kernels. The main idea lies in relating
polynomial rings to feature space, and ideals to manifolds, then exploiting
this generative-discriminative duality on kernel matrices. We illustrate this
by proposing two algorithms, IPCA and AVICA, for simultaneous manifold and
feature learning, and test their accuracy on synthetic and real world data.Comment: 15 pages, 1 figur
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