Gap Polymake Singular Applications

Polymake - Singular projects

• Normalisation of monomial ideals (C. Eder, https://github.com/ederc/Sources/tree/int-clos)

[Christian, please drop a couple of lines here]

• Computation of GIT fans (S. Keicher: Computing the GIT-fan, http://arxiv.org/abs/1205.4204) [to come after Simon's invitation to Kaiserslautern]

• Computation of Groebner fans (Bogart, Jensen, Speyer, Sturmfels, Thomas: Computing Tropical Varieties, http://arxiv.org/abs/math/0507563)

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.

• 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)

Gap - Singular projects

• Computation of the cohomology rings (Green, King: The computation of the cohomology rings of all groups of order 128, http://arxiv.org/abs/1001.2577)

possible applications (not yet thought through)

• deformation with constant Milnor number (H. Schönemann)
• computation of Weyl Algebras of line bundles over toric varieties (Cuntz, Ren, Trautmann)