6,911 research outputs found
Toric Border Bases
We extend the theory and the algorithms of Border Bases to systems of Laurent
polynomial equations, defining "toric" roots. Instead of introducing new
variables and new relations to saturate by the variable inverses, we propose a
more efficient approach which works directly with the variables and their
inverse. We show that the commutation relations and the inversion relations
characterize toric border bases. We explicitly describe the first syzygy module
associated to a toric border basis in terms of these relations. Finally, a new
border basis algorithm for Laurent polynomials is described and a proof of its
termination is given for zero-dimensional toric ideals
Ideals generated by 2-minors, collection of cells and stack polyominoes
In this paper we study ideals generated by quite general sets of 2-minors of
an -matrix of indeterminates. The sets of 2-minors are defined by
collections of cells and include 2-sided ladders. For convex collections of
cells it is shown that the attached ideal of 2-minors is a Cohen--Macaulay
prime ideal. Primality is also shown for collections of cells whose connected
components are row or column convex. Finally the class group of the ring
attached to a stack polyomino and its canonical class is computed, and a
classification of the Gorenstein stack polyominoes is given.Comment: 29 pages, 32 figure
LVMB manifolds and simplicial spheres
LVM and LVMB manifolds are a large family of examples of non kahler
manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be
seen as LVMB manifolds. The LVM manifolds have a very natural action of the
real torus and the quotient of this action is a simple polytope. This quotient
allows us to relate closely LVM manifolds to the moment-angle manifolds studied
(for example) by Buchstaber and Panov. The aim of this paper is to generalize
the polytopes associated to LVM manifolds to the LVMB case and study its
properties. In particular, we show that it belongs to a very large class of
simplicial spheres. Moreover, we show that for every sphere belonging to this
class, we can construct a LMVB manifold whose associated sphere is the given
sphere. We use this latter result to show that many moment-angle complexes can
be endowed with a complex structure (up to product with circles)
Border bases for lattice ideals
The main ingredient to construct an O-border basis of an ideal I
K[x1,. .., xn] is the order ideal O, which is a basis of the K-vector space
K[x1,. .., xn]/I. In this paper we give a procedure to find all the possible
order ideals associated with a lattice ideal IM (where M is a lattice of Z n).
The construction can be applied to ideals of any dimension (not only
zero-dimensional) and shows that the possible order ideals are always in a
finite number. For lattice ideals of positive dimension we also show that,
although a border basis is infinite, it can be defined in finite terms.
Furthermore we give an example which proves that not all border bases of a
lattice ideal come from Gr\"obner bases. Finally, we give a complete and
explicit description of all the border bases for ideals IM in case M is a
2-dimensional lattice contained in Z 2 .Comment: 25 pages, 3 figures. Comments welcome!, MEGA'2015 (Special Issue),
Jun 2015, Trento, Ital
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