On the complexity of computing Gröbner bases for weighted homogeneous systems
Abstract
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 $W=(w_{1},\dots,w_{n})$, $W$-homogeneous
polynomials are polynomials which are homogeneous w.r.t the weighted
degree
$\deg_{W}(X_{1}^{\alpha_{1}},\dots,X_{n}^{\alpha_{n}}) = \sum
w_{i}\alpha_{i}$.
Gröbner 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
$\left(\binom{n+d_{max}-1}{d_{max}}^{\omega}\right)$ can be divided by a
factor $\left( \prod w_{i} \right)^{\omega}$. For zero-dimensional
systems, the complexity of Algorithm~\FGLM $nD^{\omega}$ (where $D$
is the number of solutions of the system) can be divided by the same
factor $\left( \prod w_{i} \right)^{\omega}$.
Under genericity
assumptions, for zero-dimensional weighted homogeneous systems of
$W$-degree $(d_{1},\dots,d_{n})$, these complexity estimates are polynomial in
the weighted Bézout bound
$\prod_{i=1}^{n}d_{i} / \prod_{i=1}^{n}w_{i}$.
Furthermore, the maximum degree reached in a run of Algorithm \F5 is
bounded by the weighted Macaulay bound $\sum (d_{i}-w_{i}) + w_{n}$,
and this bound is sharp if we can order the weights so that
$w_{n}=1$. 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.
Domains
Symbolic Computation [cs.SC]Origin | Files produced by the author(s) |
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