224 research outputs found
The toric Hilbert scheme of a rank two lattice is smooth and irreducible
The toric Hilbert scheme of a lattice L in Z^n is the multigraded Hilbert
scheme parameterizing all ideals in k[x_1,...,x_n] with Hilbert function value
one for every degree in the grading monoid N^n/L. In this paper we show that if
L is two-dimensional, then the toric Hilbert scheme of L is smooth and
irreducible. This result is false for lattices of dimension three and higher as
the toric Hilbert scheme of a rank three lattice can be reducible.Comment: 19 pages, 4 figure
Convex Hulls of Algebraic Sets
This article describes a method to compute successive convex approximations
of the convex hull of a set of points in R^n that are the solutions to a system
of polynomial equations over the reals. The method relies on sums of squares of
polynomials and the dual theory of moment matrices. The main feature of the
technique is that all computations are done modulo the ideal generated by the
polynomials defining the set to the convexified. This work was motivated by
questions raised by Lov\'asz concerning extensions of the theta body of a graph
to arbitrary real algebraic varieties, and hence the relaxations described here
are called theta bodies. The convexification process can be seen as an
incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic
set in R^n. When the defining ideal is real radical the results become
especially nice. We provide several examples of the method and discuss
convergence issues. Finite convergence, especially after the first step of the
method, can be described explicitly for finite point sets.Comment: This article was written for the "Handbook of Semidefinite, Cone and
Polynomial Optimization: Theory, Algorithms, Software and Applications
Small Chvatal rank
We propose a variant of the Chvatal-Gomory procedure that will produce a
sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax
<= b} as b varies. The number of steps needed is called the small Chvatal rank
(SCR) of A. We characterize matrices for which SCR is zero via the notion of
supernormality which generalizes unimodularity. SCR is studied in the context
of the stable set problem in a graph, and we show that many of the well-known
facet normals of the stable set polytope appear in at most two rounds of our
procedure. Our results reveal a uniform hypercyclic structure behind the
normals of many complicated facet inequalities in the literature for the stable
set polytope. Lower bounds for SCR are derived both in general and for
polytopes in the unit cube.Comment: 24 pages, 3 figures, v3. Major revision: additional author, new
application to stable-set polytopes, reorganization of sections. Accepted for
publication in Mathematical Programmin
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