Residues And Resultants, 2011 University of Massachusetts - Amherst

#### Residues And Resultants, E Cattani, Alicia Dickenstein, Bernd Sturmfels

*Eduardo Cattani*

Resultants, Jacobians and residues are basic invariants of multivariate polynomial systems. We examine their interrelations in the context of toric geometry. The global residue in the torus, studied by Khovanskii, is the sum over local Grothendieck residues at the zeros of $n$ Laurent polynomials in $n$ variables. Cox introduced the related notion of the toric residue relative to $n+1$ divisors on an $n$-dimensional toric variety. We establish denominator formulas in terms of sparse resultants for both the toric residue and the global residue in the torus. A byproduct is a determinantal formula for resultants based on Jacobians.

The A-Hypergeometric System Associated With A Monomial Curve, 2011 University of Massachusetts - Amherst

#### The A-Hypergeometric System Associated With A Monomial Curve, E Cattani, C D'Andrea, A Dickenstein

*Eduardo Cattani*

No abstract provided.

Counting Solutions To Binomial Complete Intersections, 2011 University of Massachusetts - Amherst

#### Counting Solutions To Binomial Complete Intersections, E Cattani, A Dickenstein

*Eduardo Cattani*

We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.

Rational Hypergeometric Functions, 2011 University of Massachusetts - Amherst

#### Rational Hypergeometric Functions, E Cattani, A Dickenstein, B Sturmfels

*Eduardo Cattani*

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator ...

Restriction Of A-Discriminants And Dual Defect Toric Varieties, 2011 University of Massachusetts - Amherst

#### Restriction Of A-Discriminants And Dual Defect Toric Varieties, R Curran, E Cattani

*Eduardo Cattani*

We study the A-discriminant of toric varieties. We reduce its computation to the case of irreducible configurations and describe its behavior under specialization of some of the variables to zero. We give characterizations of dual defect toric varieties in terms of their Gale dual and classify dual defect toric varieties of codimension less than or equal to four.

Residues In Toric Varieties, 2011 University of Massachusetts - Amherst

#### Residues In Toric Varieties, E Cattani, D Cox, A Dickenstein

*Eduardo Cattani*

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees ...

Planar Configurations Of Lattice Vectors And Gkz-Rational Toric Fourfolds In P-6, 2011 University of Massachusetts - Amherst

#### Planar Configurations Of Lattice Vectors And Gkz-Rational Toric Fourfolds In P-6, E Cattani, A Dickenstein

*Eduardo Cattani*

We introduce a notion of balanced configurations of vectors. This is motivated by the study of rational A-hypergeometric functions in the sense of Gelfand, Kapranov and Zelevinsky. We classify balanced configurations of seven plane vectors up to GL(2,)-equivalence and deduce that the only gkz-rational toric four-folds in 6 are those varieties associated with an essential Cayley configuration. We show that in this case, all rational A-hypergeometric functions may be described in terms of toric residues. This follows from studying a suitable hyperplane arrangement.

The Structure Of Bivariate Rational Hypergeometric Functions, 2011 University of Massachusetts - Amherst

#### The Structure Of Bivariate Rational Hypergeometric Functions, E Cattani, Alicia Dickenstein, Fernando Villegas

*Eduardo Cattani*

We describe the structure of all codimension-2 lattice configurations A which admit a stable rational A-hypergeometric function, that is a rational function F all the partial derivatives of which are nonzero, and which is a solution of the A-hypergeometric system of partial differential equations defined by Gel′ fand, Kapranov, and Zelevinsky. We show, moreover, that all stable rational A-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series the coefficients of which are quotients of factorials of linear forms.

Computing Multidimensional Residues, 2011 University of Massachusetts - Amherst

#### Computing Multidimensional Residues, E Cattani, Alicia Dickenstein, Bernd Sturmfels

*Eduardo Cattani*

Given n polynomials in n variables with a finite number of complex roots, for any of their roots there is a local residue operator assigning a complex number to any polynomial. This is an algebraic, but generally not rational, function of the coefficients. On the other hand, the global residue, which is the sum of the local residues over all roots, depends rationally on the coefficients. This paper deals with symbolic algorithms for evaluating that rational function. Under the assumption that the deformation to the initial forms is flat, for some choice of weights on the variables, we express the ...

Component Groups Of Unipotent Centralizers In Good Characteristic, 2011 University of Massachusetts - Amherst

#### Component Groups Of Unipotent Centralizers In Good Characteristic, Gj Mcninch, E Sommers

*Eric N Sommers*

Let *G* be a connected, reductive group over an algebraically closed field of good characteristic. For *u**G* unipotent, we describe the conjugacy classes in the component group *A*(*u*) of the centralizer of *u*. Our results extend work of the second author done for simple, adjoint *G* over the complex numbers. When *G* is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of *A*(*u*). Thus we obtain the result, previously known through case-checking, that ...

Normality Of Very Even Nilpotent Varieties In D-2l, 2011 University of Massachusetts - Amherst

#### Normality Of Very Even Nilpotent Varieties In D-2l, E Sommers

*Eric N Sommers*

For the classical groups, Kraft and Procesi have resolved the question of which nilpotent orbits have closures that are normal and which do not, with the exception of the very even orbits in D2l that have partitions of the form (a2k, b2) for a > b even natural numbers satisfying ak + b = 2l.

B-Stable Ideals In The Nilradical Of A Borel Subalgebra, 2011 University of Massachusetts - Amherst

#### B-Stable Ideals In The Nilradical Of A Borel Subalgebra, En Sommers

*Eric N Sommers*

Let $G$G be a connected simple algebraic group over the complex numbers and $B$B a Borel subgroup. Let $\germ g$g be the Lie algebra of $G$G and $\germ b$b the Lie algebra of $B$B . A subspace of the nilradical of $\germ b$b which is stable under the action of $B$B is called a $B$B -stable ideal of the nilradical. It is called strictly positive if it intersects the simple root spaces trivially. The author counts the number of strictly positive $B$B -stable ideals in the nilradical of a Borel subalgebra ...

Exterior Powers Of The Reflection Representation In Springer Theory, 2011 University of Massachusetts - Amherst

#### Exterior Powers Of The Reflection Representation In Springer Theory, E Sommers

*Eric N Sommers*

We give a proof of a conjecture of Lehrer and Shoji regarding the occurrences of the exterior powers of the reflection representation in the cohomology of Springer fibers. The actual theorem proved is a slight extension of the original conjecture to all nilpotent orbits and also takes into account the action of the component group. The method is to use Shoji's approach to the orthogonality formulas for Green functions to relate the symmetric algebra to a sum over Green functions. In the second part of the paper we give an explanation of the appearance of the Orlik-Solomon exponents using ...

Normality Of Nilpotent Varieties In E-6, 2011 University of Massachusetts - Amherst

#### Normality Of Nilpotent Varieties In E-6, E Sommers

*Eric N Sommers*

We determine which nilpotent orbits in E6 have closures which are normal varieties and which do not. At the same time we are able to verify a conjecture in [E. Sommers, Comm. Math. Univ. Sancti Pauli 49 (1) (2000) 101–104] concerning functions on non-special nilpotent orbits for E6.

Exponents For B-Stable Ideals, 2011 University of Massachusetts - Amherst

#### Exponents For B-Stable Ideals, E Sommers, J Tymoczko

*Eric N Sommers*

Let be a simple algebraic group over the complex numbers containing a Borel subgroup . Given a -stable ideal in the nilradical of the Lie algebra of , we define natural numbers which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types and some other types. When , we recover the usual exponents of by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the ...

Local Systems On Nilpotent Orbits And Weighted Dynkin Diagrams, 2011 University of Massachusetts - Amherst

#### Local Systems On Nilpotent Orbits And Weighted Dynkin Diagrams, Promad Achar, E Sommers

*Eric N Sommers*

We study the Lusztig-Vogan bijection for the case of a local system. We compute the bijection explicitly in type A for a local system and then show that the dominant weights obtained for different local systems on the same orbit are related in a manner made precise in the paper. We also give a conjecture (putatively valid for all groups) detailing how the weighted Dynkin diagram for a nilpotent orbit in the dual Lie algebra should arise under the bijection.

Pieces Of Nilpotent Cones For Classical Groups, 2011 University of Massachusetts - Amherst

#### Pieces Of Nilpotent Cones For Classical Groups, Promad Achar, Anthony Henderson, E Sommers

*Eric N Sommers*

We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato's exotic nilpotent cone. We prove that the number of $\F_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result that corresponding special pieces in types $B_n$ and $C_n$ have the same number of $\F_q$-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent ...

Dark-Bright Discrete Solitons: A Numerical Study Of Existence, Stability And Dynamics, 2011 UMass, Amherst

#### Dark-Bright Discrete Solitons: A Numerical Study Of Existence, Stability And Dynamics, A. Alvarez, J. Cuevas, F. Romero, Panos Kevrekidis

*Panos Kevrekidis*

In the present work, we numerically explore the existence and stability properties of different types of configurations of dark-bright solitons, dark-bright soliton pairs and pairs of dark-bright and dark solitons in discrete settings, starting from the anti-continuum limit. We find that while single discrete dark-bright solitons have similar stability properties to discrete dark solitons, their pairs may only be stable if the bright components are in phase and are always unstable if the bright components are out of phase. Pairs of dark-bright solitons with dark ones have similar stability properties as individual dark or dark-bright ones. Lastly, we consider collisions ...

A New Summation Formula For Wp-Bailey Pairs, 2011 West Chester University of Pennsylvania

#### A New Summation Formula For Wp-Bailey Pairs, James Mclaughlin

*Mathematics*

No abstract provided.

Quantitative Stability Of Linear Infinite Inequality Systems Under Block Perturbations With Applications To Convex Systems, 2011 Miguel Hernández University of Elche, Alicante, Spain

#### Quantitative Stability Of Linear Infinite Inequality Systems Under Block Perturbations With Applications To Convex Systems, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra

*Mathematics Research Reports*

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is loo(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system ...