427 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
On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection
Given a zero-dimensional polynomial system consisting of n integer
polynomials in n variables, we propose a certified and complete method to
compute all complex solutions of the system as well as a corresponding
separating linear form l with coefficients of small bit size. For computing l,
we need to project the solutions into one dimension along O(n) distinct
directions but no further algebraic manipulations. The solutions are then
directly reconstructed from the considered projections. The first step is
deterministic, whereas the second step uses randomization, thus being
Las-Vegas.
The theoretical analysis of our approach shows that the overall cost for the
two problems considered above is dominated by the cost of carrying out the
projections. We also give bounds on the bit complexity of our algorithms that
are exclusively stated in terms of the number of variables, the total degree
and the bitsize of the input polynomials
A survey on signature-based Gr\"obner basis computations
This paper is a survey on the area of signature-based Gr\"obner basis
algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain
the general ideas behind the usage of signatures. We show how to classify the
various known variants by 3 different orderings. For this we give translations
between different notations and show that besides notations many approaches are
just the same. Moreover, we give a general description of how the idea of
signatures is quite natural when performing the reduction process using linear
algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table
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