3,715 research outputs found
A non-regular Groebner fan
The Groebner fan of an ideal , defined by Mora and
Robbiano, is a complex of polyhedral cones in . The maximal cones of the
fan are in bijection with the distinct monomial initial ideals of as the
term order varies. If is homogeneous the Groebner fan is complete and is
the normal fan of the state polytope of . In general the Groebner fan is not
complete and therefore not the normal fan of a polytope. We may ask if the
restricted Groebner fan, a subdivision of , is regular i.e. the
normal fan of a polyhedron. The main result of this paper is an example of an
ideal in whose restricted Groebner fan is not regular.Comment: 11 page
Computing Groebner Fans
This paper presents algorithms for computing the Groebner fan of an arbitrary
polynomial ideal. The computation involves enumeration of all reduced Groebner
bases of the ideal. Our algorithms are based on a uniform definition of the
Groebner fan that applies to both homogeneous and non-homogeneous ideals and a
proof that this object is a polyhedral complex. We show that the cells of a
Groebner fan can easily be oriented acyclically and with a unique sink,
allowing their enumeration by the memory-less reverse search procedure. The
significance of this follows from the fact that Groebner fans are not always
normal fans of polyhedra in which case reverse search applies automatically.
Computational results using our implementation of these algorithms in the
software package Gfan are included.Comment: 26 page
The Circuit Ideal of a Vector Configuration
The circuit ideal, \ica, of a configuration \A = \{\a_1, ..., \a_n\}
\subset \Z^d is the ideal generated by the binomials {\x}^{\cc^+} -
{\x}^{\cc^-} \in \k[x_1, ..., x_n] as \cc = \cc^+ - \cc^- \in \Z^n varies
over the circuits of \A. This ideal is contained in the toric ideal, \ia,
of \A which has numerous applications and is nontrivial to compute. Since
circuits can be computed using linear algebra and the two ideals often
coincide, it is worthwhile to understand when equality occurs. In this paper we
study \ica in relation to \ia from various algebraic and combinatorial
perspectives. We prove that the obstruction to equality of the ideals is the
existence of certain polytopes. This result is based on a complete
characterization of the standard pairs/associated primes of a monomial initial
ideal of \ica and their differences from those for the corresponding toric
initial ideal. Eisenbud and Sturmfels proved that \ia is the unique minimal
prime of \ica and that the embedded primes of \ica are indexed by certain
faces of the cone spanned by \A. We provide a necessary condition for a
particular face to index an embedded prime and a partial converse. Finally, we
compare various polyhedral fans associated to \ia and \ica. The Gr\"obner
fan of \ica is shown to refine that of \ia when the codimension of the
ideals is at most two.Comment: 25 page
Anomalous Non-Hydrogenic Exciton Series in 2D Materials on High- Dielectric Substrates
Engineering of the dielectric environment represents a powerful strategy to
control the electronic and optical properties of two-dimensional (2D) materials
without compromising their structural integrity. Here we show that the recent
development of high- 2D materials present new opportunities for
dielectric engineering. By solving a 2D Mott-Wannier exciton model for WSe
on different substrates using a screened electron-hole interaction obtained
from first principles, we demonstrate that the exciton Rydberg series changes
qualitatively when the dielectric screening within the 2D semiconductor becomes
dominated by the substrate. In this regime, the distance dependence of the
screening is reversed and the effective screening increases with exciton
radius, which is opposite to the conventional 2D screening regime.
Consequently, higher excitonic states become underbound rather than overbound
as compared to the Hydrogenic Rydberg series. Finally, we derive a general
analytical expression for the exciton binding energy of the entire 2D Rydberg
serie
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