3 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
Multigraded Hilbert Functions and Buchberger Algorithm
In this paper we continue the work done in [GMRT]. Namely we describe an algorithm which allows to drive the classical Buchberger Algorithm by means of multigraded Hilbert-Poincare series. The result is that in the multigraded case one can kill many more useless critical pairs. Besides, the new algorithm, calLed MGHDriven, is intrinsically parallelizable and it is now implemented in CoCoA (see [CNR])