- Graded commutative algebra and related structures in Singular with applications (2011)
- This thesis is devoted to constructive module theory of polynomial graded commutative algebras over a field. It treats the theory of Groebner bases (GB), standard bases (SB) and syzygies as well as algorithms and their implementations. Graded commutative algebras naturally unify exterior and commutative polynomial algebras. They are graded non-commutative, associative unital algebras over fields and may contain zero-divisors. In this thesis we try to make the most use out of _a priori_ knowledge about their characteristic (super-commutative) structure in developing direct symbolic methods, algorithms and implementations, which are intrinsic to graded commutative algebras and practically efficient. For our symbolic treatment we represent them as polynomial algebras and redefine the product rule in order to allow super-commutative structures and, in particular, to allow zero-divisors. Using this representation we give a nice characterization of a GB and an algorithm for its computation. We can also tackle central localizations of graded commutative algebras by allowing commutative variables to be _local_, generalizing Mora algorithm (in a similar fashion as G.M.Greuel and G.Pfister by allowing local or mixed monomial orderings) and working with SBs. In this general setting we prove a generalized Buchberger's criterion, which shows that syzygies of leading terms play the utmost important role in SB and syzygy module computations. Furthermore, we develop a variation of the La Scala-Stillman free resolution algorithm, which we can formulate particularly close to our implementation. On the implementation side we have further developed the Singular non-commutative subsystem Plural in order to allow polynomial arithmetic and more involved non-commutative basic Computer Algebra computations (e.g. S-polynomial, GB) to be easily implementable for specific algebras. At the moment graded commutative algebra-related algorithms are implemented in this framework. Benchmarks show that our new algorithms and implementation are practically efficient. The developed framework has a lot of applications in various branches of mathematics and theoretical physics. They include computation of sheaf cohomology, coordinate-free verification of affine geometry theorems and computation of cohomology rings of p-groups, which are partially described in this thesis.
- Computation of the central elements and centralizers of sets of elements in non-commutative polynomial algebras (2006)
- In this thesis we present the implementation of libraries center.lib and perron.lib for the non-commutative extension Plural of the Computer Algebra System Singular. The library center.lib was designed for the computation of elements of the centralizer of a set of elements and the center of a non-commutative polynomial algebra. It also provides solutions to related problems. The library perron.lib contains a procedure for the computation of relations between a set of pairwise commuting polynomials. The thesis comprises the theory behind the libraries, aspects of the implementation and some applications of the developed algorithms. Moreover, we provide extensive benchmarks for the computation of elements of the center. Some of our examples were never computed before.