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
  Abstract:
  Implementation: currently Gfan computes the Groebner fan for polynomial ideals over rationals, data stored in gfanlib; possible complete extention to gfanlib by Anders, reuse M. Monerjan's code for Groebner fan computation of polynomial ideal over more general ground fields.
  System: Singular with polymake.so
  References: Bogart, Jensen, Speyer, Sturmfels, Thomas: Computing Tropical Varieties, http://arxiv.org/abs/math/0507563

• Computation of tropical varieties
  (Bogart, Jensen, Speyer, Sturmfels, Thomas: Computing Tropical Varieties, http://arxiv.org/abs/math/0507563)
  currently Gfan computes the tropcial variety for polynomial ideals over rationals, data stored in gfanlib; possible complete extention to gfanlib by Anders, direct implementation of the algorithm in Singular for more general polynomial rings (possibly over valued fields)

• Framework for polyhedral divisors
  (Altmann, Hausen: Polyhedral Divisors and Algebraic Torus Actions, http://arxiv.org/abs/math/0306285)

• Generators of multigraded algebras
  (Ilten, Kastner: Generators of multigraded algebras, http://arxiv.org/abs/1203.5382)

• Regularity of semigroup algebras
  (Boehm, Eisenbud, Nitsche: Decomposition of semigroup algebras, http://arxiv.org/abs/1110.3653)

• Ring theoretic properties of semigroup algebras
  (Boehm, Eisenbud, Nitsche: Decomposition of Monomial Algebras: Applications and Algorithms, http://arxiv.org/abs/1206.1735)

• Deformations with constant Milnor number
  People: H. Schönemann, J. Boehm

• Computation of Weyl Algebras of line bundles over toric varieties
  People: M. Cuntz, Y. Ren, G. Trautmann
  Abstract:
  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 could redo the sage implementation using GAP and Singular/Plural for the Groebner computations
  Abstract:
  References: Green, King: The computation of the cohomology rings of all groups of order 128, http://arxiv.org/abs/1001.2577