Letterplace

We present the newest release of the subsystem of Singular called Letterplace which exists since 2009. It is devoted to computations with finitely presented associative algebras over fields and offers Gröbner(-Shirshov) bases over free algebras via the Letterplace correspondence of La Scala and Levandovskyy. This allows to use highly tuned commutative data structures internally and to reuse parts of existing algorithms in the non-commutative situation. The present version has been deeply reengineered, based on the experience with earlier and experimental versions. We offer an unprecedented functionality, some of which for the first time in the history of computer algebra. In particular, we present tools for elimination theory (via truncated Gröbner bases and via supporting several kinds of elimination orderings), dimension theory (Gel'fand-Kirillov and global dimension), and for homological algebra (such as syzygy bimodules and lifts for ideals and bimodules) to name a few. Another article in this issue is devoted to the extension of Gröbner bases to the coefficients in principal ideal rings including Z, which is also a part of this release. We report on comparison with other systems and on some advances in the theory. Quite nontrivial examples illustrate the abilities of the system.