112 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
Predicting zero reductions in Gr\"obner basis computations
Since Buchberger's initial algorithm for computing Gr\"obner bases in 1965
many attempts have been taken to detect zero reductions in advance.
Buchberger's Product and Chain criteria may be known the most, especially in
the installaton of Gebauer and M\"oller. A relatively new approach are
signature-based criteria which were first used in Faug\`ere's F5 algorithm in
2002. For regular input sequences these criteria are known to compute no zero
reduction at all. In this paper we give a detailed discussion on zero
reductions and the corresponding syzygies. We explain how the different methods
to predict them compare to each other and show advantages and drawbacks in
theory and practice. With this a new insight into algebraic structures
underlying Gr\"obner bases and their computations might be achieved.Comment: 25 pages, 3 figure
A Borel open cover of the Hilbert scheme
Let be an admissible Hilbert polynomial in \PP^n of degree . The
Hilbert scheme \hilb^n_p(t) can be realized as a closed subscheme of a
suitable Grassmannian , hence it could be globally defined by
homogeneous equations in the Plucker coordinates of and covered by
open subsets given by the non-vanishing of a Plucker coordinate, each embedded
as a closed subscheme of the affine space , . However,
the number of Plucker coordinates is so large that effective computations
in this setting are practically impossible. In this paper, taking advantage of
the symmetries of \hilb^n_p(t), we exhibit a new open cover, consisting of
marked schemes over Borel-fixed ideals, whose number is significantly smaller
than . Exploiting the properties of marked schemes, we prove that these open
subsets are defined by equations of degree in their natural
embedding in \Af^D. Furthermore we find new embeddings in affine spaces of
far lower dimension than , and characterize those that are still defined by
equations of degree . The proofs are constructive and use a
polynomial reduction process, similar to the one for Grobner bases, but are
term order free. In this new setting, we can achieve explicit computations in
many non-trivial cases.Comment: 17 pages. This version contains and extends the first part of version
2 (arXiv:0909.2184v2[math.AG]). A new extended version of the second part,
with some new results, is posed at arxiv:1110.0698v3[math.AC]. The title is
slightly changed. Final version accepted for publicatio
Representations of fundamental groups of 3-manifolds into PGL(3,C): Exact computations in low complexity
In this paper we are interested in computing representations of the
fundamental group of a 3-manifold into PSL(3;C) (in particular in PSL(2;C);
PSL(3;R) and PU(2; 1)). The representations are obtained by gluing decorated
tetrahedra of flags. We list complete computations (giving 0-dimensional or
1-dimensional solution sets) for the first complete hyperbolic non-compact
manifolds with finite volume which are obtained gluing less than three
tetrahedra with a description of the computer methods used to find them
Moment ideals of local Dirac mixtures
In this paper we study ideals arising from moments of local Dirac measures
and their mixtures. We provide generators for the case of first order local
Diracs and explain how to obtain the moment ideal of the Pareto distribution
from them. We then use elimination theory and Prony's method for parameter
estimation of finite mixtures. Our results are showcased with applications in
signal processing and statistics. We highlight the natural connections to
algebraic statistics, combinatorics and applications in analysis throughout the
paper.Comment: 26 pages, 3 figure
Axioms for a theory of signature bases
Twenty years after the discovery of the F5 algorithm, Gr\"obner bases with
signatures are still challenging to understand and to adapt to different
settings. This contrasts with Buchberger's algorithm, which we can bend in many
directions keeping correctness and termination obvious. I propose an axiomatic
approach to Gr\"obner bases with signatures with the purpose of uncoupling the
theory and the algorithms, and giving general results applicable in many
different settings (e.g. Gr\"obner for submodules, F4-style reduction,
noncommutative rings, non-Noetherian settings, etc.)
A new approach based on quadratic forms to attack the McEliece cryptosystem
We bring in here a novel algebraic approach for attacking the McEliece
cryptosystem. It consists in introducing a subspace of matrices representing
quadratic forms. Those are associated with quadratic relationships for the
component-wise product in the dual of the code used in the cryptosystem.
Depending on the characteristic of the code field, this space of matrices
consists only of symmetric matrices or skew-symmetric matrices. This matrix
space is shown to contain unusually low-rank matrices (rank or
depending on the characteristic) which reveal the secret polynomial structure
of the code. Finding such matrices can then be used to recover the secret key
of the scheme. We devise a dedicated approach in characteristic consisting
in using a Gr\"obner basis modeling that a skew-symmetric matrix is of rank
. This allows to analyze the complexity of solving the corresponding
algebraic system with Gr\"obner bases techniques. This computation behaves
differently when applied to the skew-symmetric matrix space associated with a
random code rather than with a Goppa or an alternant code. This gives a
distinguisher of the latter code family. We give a bound on its complexity
which turns out to interpolate nicely between polynomial and exponential
depending on the code parameters. A distinguisher for alternant/Goppa codes was
already known [FGO+11]. It is of polynomial complexity but works only in a
narrow parameter regime. This new distinguisher is also polynomial for the
parameter regime necessary for [FGO+11] but contrarily to the previous one is
able to operate for virtually all code parameters relevant to cryptography.
Moreover, we use this matrix space to find a polynomial time attack of the
McEliece cryptosystem provided that the Goppa code is distinguishable by the
method of [FGO+11] and its degree is less than , where is the alphabet
size of the code.Comment: 61 page
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