State Polytope Research Articles

State polytopes related to two classes of combinatorial neural codes

Combinatorial neural codes are collections of 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn on the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called 2-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality, part of the difficulty coming from the large number of codewords possible when additional neurons are used.In this article, we study two infinite classes of combinatorial neural codes in detail. For each code, we explicitly compute its universal Gröbner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

Generic semistability for reductive group actions

We prove that given a reductive algebraic group G G and a rational representation ρ : G → G L ( V ) \rho : G \to \mathrm {GL}(V) defined over an algebraically closed field of characteristic 0 0 , v ∈ V v \in V is generically semistable, i.e., 0 ∉ T . v ¯ 0 \not \in \overline {T.v} for a general maximal torus T T if and only if v v is semistable with respect to the induced action of the center of G G . The proof is obtained through a detailed description of the relation between the state polytope with respect to the maximal torus T T of G G and the state polytope with respect to T ∩ [ G , G ] T \cap [G,G] . We also consider the case of solvable groups and prove that the generic semistability implies the center semistability but not the other way around.

A State Polytope Decomposition Formula

AbstractWe give a decomposition formula for computing the state polytope of a reducible variety in terms of the state polytopes of its components: if a polarized projective variety X is a chain of subvarieties Xi satisfying some further conditions, then the state polytope of X is the Minkowski sum of the state polytopes of Xi translated by a vector τ, which can be readily computed from the ideal of Xi. The decomposition is in the strongest sense in that the vertices of the state polytope of X are precisely the sum of vertices of the state polytopes of Xi translated by τ. We also give a similar decomposition formula for the Hilbert–Mumford index of the Hilbert points of X. We give a few examples of the state polytope and the Hilbert–Mumford index computation of reducible curves, which are interesting in the context of the log minimal model program for the moduli space of stable curves.

Two-Factor Saturated Designs: Cycles, Gini Index, and State Polytopes

In this article, we analyze and characterize the saturated fractions of two-factor designs under the simple effect model. Using linear algebra, we define a criterion to check whether a given fraction is saturated or not. We also compute the number of saturated fractions, providing an alternative proof of the Cayley’s formula. Finally, we show how, given a list of saturated fractions, Gini indexes of their margins and the associated state polytopes could be used to classify them.

Control to Facet by Piecewise-Affine Output Feedback

The control-to-facet problem plays an important role in the design of feedback controllers for piecewise-affine hybrid systems on polytopes. In the literature, necessary conditions and sufficient conditions for solvability by static state feedback exist. In this paper, we extend these results to the case of continuous piecewise-affine static output feedback. For the construction of a controller, a triangulation of the output polytope is made which satisfies additional conditions to guarantee compatibility with the induced subdivision of the state polytope. In the state feedback case, the use of this special type of triangulation is not required.

Minimal average degree aberration and the state polytope for experimental designs

For a particular experimental design, there is interest in finding which polynomial models can be identified in the usual regression set up. The algebraic methods based on Grobner bases provide a systematic way of doing this. The algebraic method does not, in general, produce all estimable models but it can be shown that it yields models which have minimal average degree in a well-defined sense and in both a weighted and unweighted version. This provides an alternative measure to that based on “aberration” and moreover is applicable to any experimental design. A simple algorithm is given and bounds are derived for the criteria, which may be used to give asymptotic Nyquist-like estimability rates as model and sample sizes increase.

Polyhedra: A package for computations with convex polyhedral objects

Convex polyhedral objects play an important role in numerous areas of algebraic geometry. The package Polyhedra enables one to work with such convex polyhedral objects in Macaulay 2. In particular, Polyhedra implements objects representing polyhedra, cones, and fans. Many standard operations, such as intersections, duality, convex hulls, and calculation of Hilbert bases are included. The package also provides functions to create convex polyhedral objects from some algebraic objects. THE PACKAGE AND ITS APPLICATIONS. Many objects in algebraic geometry can be described by convex polyhedral objects. For example, the Grobner degenerations of a homogeneous ideal are in one-to-one correspondence with the faces of its state polytope; see (St, Chapter 2). Geometric invariant theory (GIT) quotients coming from linearizations of the trivial bundle provide a second class of examples. For an effective action of the torus (C ) k on affine space A n , there is an associated exact sequence of lattices 0! K! Z n ! Z k ! 0. The GIT-quotients of this action are given by the normal fans of certain polyhedra in the associated Q-vector space KQ; for more information see (B, §III.2). The package Polyhedra provides methods within Macaulay 2 (M2) for creating and manipulating convex polyhedral objects such as polyhedra and cones. Some objects from algebraic geometry can also be translated into their corresponding convex polyhedral objects.

Lectures in geometric combinatorics

Abstract algebra: Groups, rings and fields Convex polytopes: Definitions and examples Faces of polytopes Schlegel diagrams Gale diagrams Bizarre polytopes Triangulations of point configurations The secondary polytope The permutahedron Abstract algebra: Polynomial rings Grobner bases I Grobner bases II Initial complexes of toric ideals State polytopes of toric ideals Bibliography Index.

A Non-Regular Grobner Fan

The Groebner fan of an ideal $I\subset k[x_1,...,x_n]$, defined by Mora and Robbiano, is a complex of polyhedral cones in $R^n$. The maximal cones of the fan are in bijection with the distinct monomial initial ideals of $I$ as the term order varies. If $I$ is homogeneous the Groebner fan is complete and is the normal fan of the state polytope of $I$. 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 $R_{>=0}^n$, is regular i.e. the normal fan of a polyhedron. The main result of this paper is an example of an ideal in $Q[x_1,...,x_4]$ whose restricted Groebner fan is not regular.

Computing Gröbner Fans of Toric Ideals

The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Gröbner fan of the ideal is the normal fan of its state polytope. In this paper we present a software system called TiGERS (Toric Gröbner basesEnumeration by Reverse Search) for computing the Gröbner fan of a toric ideal by enumerating the edge graph of its state polytope. The key contributions are an inexpensive algorithm for local change of Gröbner bases in toric ideals and the identification of a reverse search tree on the vertices of the state polytope. Using these ideas we obtain a combinatorial Grabner walk procedure for toric ideals. TiGERS has been used to compute state polytopes with over 200,000 vertices.

Gröbner bases and triangulations of the second hypersimplex

The algebraic technique of Grobner bases is applied to study triangulations of the second hypersimplex Δ(2,n). We present a quadratic Grobner basis for the associated toric idealK(K n ). The simplices in the resulting triangulation of Δ(2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK n . Forn≥6 the number of distinct initial ideals ofI(K n ) exceeds the number of regular triangulations of Δ(2,n); more precisely, the secondary polytope of Δ(2,n) equals the state polytope ofI(K n ) forn≤5 but not forn≥6. We also construct a non-regular triangulation of Δ(2,n) forn≥9. We determine an explicit universal Grobner basis ofI(K n ) forn≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated.

Standard bases and geometric invariant theoryI. Initial ideals and state polytopes

We characterise the set of monomial ideals that can occur as the initial ideals of a given homogeneous ideal Q in a polynomial ring, as one varies the monomial order used within a fixed coordinate system. This set is canonically bijective to the set of vertices of the state polytope of Q , a convex polytope arising in the geometric invariant theory of the Hilbert point of Q .