16,995 research outputs found
An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes
We develop an incremental algorithm to compute the Newton polytope
of the resultant, aka resultant polytope, or its
projection along a given direction.
The resultant is fundamental in algebraic elimination and
in implicitization of parametric hypersurfaces.
Our algorithm exactly computes vertex- and halfspace-representations
of the desired polytope using an oracle producing resultant vertices in a
given direction.
It is output-sensitive as it uses one oracle call per vertex.
We overcome the bottleneck of determinantal predicates
by hashing, thus accelerating execution from to times.
We implement our algorithm using the experimental CGAL package {\tt
triangulation}.
A variant of the algorithm computes successively tighter inner and outer
approximations: when these polytopes have, respectively,
90\% and 105\% of the true volume, runtime is reduced up to times.
Our method computes instances of -, - or -dimensional polytopes
with K, K or vertices, resp., within hr.
Compared to tropical geometry software, ours is faster up to
dimension or , and competitive in higher dimensions
Towards Mixed Gr{\"o}bner Basis Algorithms: the Multihomogeneous and Sparse Case
One of the biggest open problems in computational algebra is the design of
efficient algorithms for Gr{\"o}bner basis computations that take into account
the sparsity of the input polynomials. We can perform such computations in the
case of unmixed polynomial systems, that is systems with polynomials having the
same support, using the approach of Faug{\`e}re, Spaenlehauer, and Svartz
[ISSAC'14]. We present two algorithms for sparse Gr{\"o}bner bases computations
for mixed systems. The first one computes with mixed sparse systems and
exploits the supports of the polynomials. Under regularity assumptions, it
performs no reductions to zero. For mixed, square, and 0-dimensional
multihomogeneous polynomial systems, we present a dedicated, and potentially
more efficient, algorithm that exploits different algebraic properties that
performs no reduction to zero. We give an explicit bound for the maximal degree
appearing in the computations
Elimination for generic sparse polynomial systems
We present a new probabilistic symbolic algorithm that, given a variety
defined in an n-dimensional affine space by a generic sparse system with fixed
supports, computes the Zariski closure of its projection to an l-dimensional
coordinate affine space with l < n. The complexity of the algorithm depends
polynomially on combinatorial invariants associated to the supports.Comment: 22 page
Toric Generalized Characteristic Polynomials
We illustrate an efficient new method for handling polynomial systems with
degenerate solution sets. In particular, a corollary of our techniques is a new
algorithm to find an isolated point in every excess component of the zero set
(over an algebraically closed field) of any by system of polynomial
equations. Since we use the sparse resultant, we thus obtain complexity bounds
(for converting any input polynomial system into a multilinear factorization
problem) which are close to cubic in the degree of the underlying variety --
significantly better than previous bounds which were pseudo-polynomial in the
classical B\'ezout bound. By carefully taking into account the underlying toric
geometry, we are also able to improve the reliability of certain sparse
resultant based algorithms for polynomial system solving
Toric Intersection Theory for Affine Root Counting
Given any polynomial system with fixed monomial term structure, we give
explicit formulae for the generic number of roots with specified coordinate
vanishing restrictions. For the case of affine space minus an arbitrary union
of coordinate hyperplanes, these formulae are also the tightest possible upper
bounds on the number of isolated roots. We also characterize, in terms of
sparse resultants, precisely when these upper bounds are attained. Finally, we
reformulate and extend some of the prior combinatorial results of the author on
which subsets of coefficients must be chosen generically for our formulae to be
exact.
Our underlying framework provides a new toric variety setting for
computational intersection theory in affine space minus an arbitrary union of
coordinate hyperplanes. We thus show that, at least for root counting, it is
better to work in a naturally associated toric compactification instead of
always resorting to products of projective spaces
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