# Gap Polymake Singular Applications

## Polymake - Singular projects

• Normalization of monomial ideals
People: Christian Eder.
Abstract: Computing the normalization of an ideal is a hard task in general. Right now the only useful algorithm known makes a detour over the Rees algebra. For the special situation of monomial ideals one can compute the integral closure resp. normalization using plain geometry: The exponent set of the integral closure of a monomial ideal I equals all the integer lattice points in the convex hull of the exponent set of I ( see Proposition 1.4.6 in the Swanson/Huneke reference below ).
Implementation: We select the exponent vectors of the given monomial ideal I , adjusts the dimension by adding vectors (1,0,...,0) up to (0,...,0,1,0) of dimension (number variables + 1) and create the corresponding cone of these points (those vectors are needed to embed the cone correctly). We get the exponent vectors for the integral closure of I by extracting the points in the corresponding Hilbert basis of the cone.
System: Singular with polymake.so
References: C. Eder, https://github.com/ederc/Sources/tree/int-clos
I. Swanson / G. Huneke, http://people.reed.edu/~iswanson/book/index.html

• Computation of GIT fans
People: Simon Keicher (will visit 15.10-19.10.), Janko Boehm, Yue Ren
Abstract: Using Groebner bases and polyhedral techniques an algorithm is given to compute the GIT-fan of algebraic torus actions on affine varieties. The GIT-fan is a more coarse (hence more useable) version of the GKZ-fan.
Implementation: The plan is to invite Simon to Kaiserslautern and to convert Simon's implementation, which uses Maple/convex and Groebner bases in Maple to Singular+Polymake. Simon, please put more some details.
System: Singular with polymake.so
References: S. Keicher: Computing the GIT-fan, http://arxiv.org/abs/1205.4204)

• Computation of Groebner fans
People: Janko Boehm, Anders Jensen, Thomas Markwig, Martin Monerjan, Yue Ren
Abstract: Bases on a definition of the Groebner fan that applies to both homogeneous and non-homogeneous ideals the Groebner fan can be computed by enumerating all reduced Groebner bases in a reverse search procedure.
Implementation: The goal is to provide a native Singular+Polymake implementation. Reuse M. Monerjan's code for Groebner fan computation of polynomial ideal over more general ground fields.
For the moment, there is a new interface package gfaninterface.lib which calls Gfan to compute the Groebner fan for polynomial ideals over prime fields and stores the data in the fan/cone structure provided by gfanlib in Singular.
extend the functionality of gfanlib by Anders.
System: Singular with polymake.so
References: Fukuda, Jensen, Thomas: Computing Groebner Fans, http://arxiv.org/abs/math.AC/0509544
Bogart, Jensen, Speyer, Sturmfels, Thomas: Computing Tropical Varieties, http://arxiv.org/abs/math/0507563

• Computation of tropical varieties
People: Janko Boehm, Anders Jensen, Thomas Markwig, Martin Monerjan, Yue Ren
Abstract: The tropical variety of a d-dimensional prime ideal in a polynomial ring with complex coefficients is a pure d-dimensional polyhedral fan connected in codimension one. It sits in the space of weight orderings and consists of those faces of the Groebner fan such that the corresponding initial ideals do not contain a monomial.
Implementation: The goal is to provide a native Singular+Polymake implementation (possibly over valued fields). Reuse M. Monerjan's code for Groebner fan computation of polynomial ideal over more general ground fields.
For the moment, there is a new interface package gfaninterface.lib which calls Gfan to compute the tropical variety for polynomial ideals over prime fields and stores the data in the fan/cone structure provided by gfanlib in Singular.
Extend the functionality of gfanlib by Anders.
System: Singular with polymake.so, Singular interfacing to gfanlib
References: Bogart, Jensen, Speyer, Sturmfels, Thomas: Computing Tropical Varieties, http://arxiv.org/abs/math/0507563)

• Framework for polyhedral divisors
People: Janko Boehm, Lars Kastner, Benjamin Lorenz, Yue Ren, Hans Schoenemann.
Abstract: By the work of Altmann and Hausen, normal affine varieties with effective algebraic torus action can be described in terms of proper polyhedral divisors on semiprojective varieties. This extends the classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.
Implementation: The basic framework is available in the library divisors.lib
System: Singular with polymake.so
References: Altmann, Hausen: Polyhedral Divisors and Algebraic Torus Actions, http://arxiv.org/abs/math/0306285)

People: Janko Boehm, Lars Kastner, Yue Ren.
Abstract: Based on the algorithm to find generators of the multigraded algebra A associated to an arbitrary p-divisor D on some variety Y, we compute generators for the Cox ring of the smooth del Pezzo surface of degree 5.
Implementation: Singular program using divisors.lib
System: Singular with polymake.so
References: Ilten, Kastner: Generators of multigraded algebras, http://arxiv.org/abs/1203.5382

• Regularity of semigroup algebras
People: Janko Boehm
Abstract: Let A in B be cancellative abelian semigroups, and let R be an integral domain. The semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings there is an algorithm computing the decomposition. In particular this gives a fast algorithm to compute the Castelnuovo-Mumford regularity of homogeneous semigroup rings.
Implementation: Convert the Macaulay2 implementation to Singular to use polymake.so for the polyhedral computations and parallel.lib to test the Eisenbud-Goto conjecture in new cases. The basic Singular implementation has been done together with Marcel Mueller.
References: Boehm, Eisenbud, Nitsche: Decomposition of semigroup algebras, http://arxiv.org/abs/1110.3653

• Ring theoretic properties of semigroup algebras
People: Janko Boehm
Abstract: Let A in B be cancellative abelian semigroups, and let R be an integral domain. The semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings there is an algorithm computing the decomposition. Based on the decomposition one can give algorithms to test ring-theoretic properties of simplicial semigroup rings like Buchsbaum, Cohen-Macaulay, Gorenstein, normal, and seminormal (all of which imply the Eisenbud-Goto conjecture).
Implementation: Convert the Macaulay2 implementation to Singular to use polymake.so for the polyhedral computations and parallel.lib to test the Eisenbud-Goto conjecture in new cases. The basic Singular implementation of the decomposition has been done together with Marcel Mueller.
References: Boehm, Eisenbud, Nitsche: Decomposition of Monomial Algebras: Applications and Algorithms, http://arxiv.org/abs/1206.1735

• Deformations with constant Milnor number
People: Hans Schönemann, Janko Boehm
Abstract: Hans, please put some lines on the mathematical background here
Implementation: Hans, please put some lines on the specific algorithm here.
System: Singular with polymake.so
References:

• Computation of Weyl Algebras of line bundles over toric varieties:
People: Michael Cuntz, Yue Ren, Guenther Trautmann
Abstract: The Nichols algebras of finite dimension over abelian groups in the complex numbers were classified by Istvan Heckenberger in the years 2004–2005 by classifying arithmetic root systems and generalized Dynkin diagrams. In more general fields, the symmetry group of the root system is replaced by a Weyl-groupoid instead. While finite Weyl-groupoid have been completely classified in a series of papers by M. Cuntz and I. Heckenberger, it is not clear whether each groupoid does arise from a Nichols algebra. Understanding the Weyl-groupoid in terms of toric geometry might give hints on such a way back.
Implementation: [algorithms to come]
System: Singular with polymake.so
References: Cunz, Ren, Trautmann: Strongly symmetric smooth toric varieties, http://arxiv.org/abs/1108.1886

##Gap - Singular projects

• Homalg project
People: Mohamed Barakat et al.
Abstract: he homalg project is a multi-author multi-package open source software project for constructive homological algebra. Although the central part of the source code is the formalization of abstract notions like Abelian categories, our focus lies on concrete applications ranging from linear control theory to commutative algebra and algebraic geometry.
Implementation: Mainly written in GAP4, the homalg project allows the use of other CASs for specific time ciritical tasks. Based on its abstract implementation of Abelian categories, it provides a wide range of specific package. It is capable to calculate resolutions of graded modules over multigraded rings, and examine sheaves over projective spaces and toric varieties represented by their Cox rings and fans. Furthermore it provides algorithms for equivariant sheaves over projective spaces and control theory.
A main part are learning objects, which store their already computed properties and use implemented theorems to deduce further properties as early as possible. It also offers a set of tools for implementation. Those are constructors for functorial methods, and a modular knowledge spreading system, so called to-do-lists.
Very soon there will also be abstract objects implemented, for example a module over a ring, only given with properties, but without any representation. Those objects will be capable to proof theorems and display their proofs using to-do-lists.
System: GAP interfacing to Singular, Polymake and other CASs.
References:
http://homalg.math.rwth-aachen.de/
Barakat, Mohamed and Lange-Hegermann, Markus, An Axiomatic Setup For Algorithmic Homological Algebra and an Alternative Approach to Localization, J. Algebra Appl. (http://arXiv.org/abs/1003.1943).

• Computation of cohomology rings
People: Probably a student.
Abstract: Computation of modular group cohomology is, essentially, a functor mapping a group homomorphism (represented in GAP) to a cohomology ring (represented in Singular).
Implemenation: Redo the sage implementation using GAP and Singular/Plural for the Groebner computations.
System: GAP with Singular interface. Or just use Singular for the Groebner computations. However this uses an F5-style linear algebra algorithm (discuss with Christian Eder).
References: Green, King: The computation of the cohomology rings of all groups of order 128, http://arxiv.org/abs/1001.2577

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