Gap Polymake Singular Applications

Polymake - Singular projects

• Normalization of monomial ideals
  People: Christian Eder.
  Abstract: Christian, please drop a couple of lines on the theory here.
  Implementation: Please comment on how Polymake is used and what is the benefit.
  System: Singular with polymake.so
  References: C. Eder, https://github.com/ederc/Sources/tree/int-clos
  More references?

• Computation of GIT fans
  People: Simon Keicher (to be contacted), 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. Simon, please put some more lines on the mathematical background of the algorithm and possible future developments.
  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)
  

• Generators of multigraded algebras
  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: Yue, please put some lines on the mathematical background here
  Implementation: Yue, please put some lines on the specific algorithm here.
  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
  Abstract: please drop some lines on the mathematical background here
  Implementation: Please write some short lines what is available, how it is implemented and what is planned.
  System: GAP interfacing to Singular
  References: http://homalg.math.rwth-aachen.de/

• 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