Enhanced computations of gröbner bases in free algebras as a new application of the letterplace paradigm

Abstract

Recently, the notion of "letterplace correspondence" between ideals in the free associative algebra KX and certain ideals in the so-called letterplace ring KXP has evolved. We continue this research direction, started by La Scala and Levandovskyy, and present novel ideas, supported by the implementation, for effective computations with ideals in the free algebra by utilizing the generalized letterplace correspondance. In particular, we provide a direct algorithm to compute Gröbner bases of non-graded ideals. Surprizingly we realize its behavior as "homogenizing without a homogenization variable". Moreover, we develop new shift-invariant data structures for this family of algorithms and discuss about them. Furthermore we generalize the famous criteria of Gebauer-Möller to the non-commutative setting and show the benefits for the computation by allowing to skip unnecessary critical pairs. The methods are implemented in the computer algebra system Singular. We present a comparison of performance of our implementation with the corresponding implementations in the systems Magma [BCP97] and GAP [GAP13] on the representative set of nontrivial examples.