 
Summary: DEGREE BOUNDS FOR GR šOBNER BASES IN ALGEBRAS OF SOLVABLE
TYPE
MATTHIAS ASCHENBRENNER AND ANTON LEYKIN
ABSTRACT. We establish doublyexponential degree bounds for Gršobner bases in certain
algebras of solvable type over a field (as introduced by KandriRody and Weispfenning).
The class of algebras considered here includes commutative polynomial rings, Weyl alge
bras, and universal enveloping algebras of finitedimensional Lie algebras. For the com
putation of these bounds, we adapt a method due to DubŽe based on a generalization of
Stanley decompositions. Our bounds yield doublyexponential degree bounds for ideal
membership and syzygies, generalizing the classical results of Hermann and Seidenberg
(in the commutative case) and Grigoriev (in the case of Weyl algebras).
INTRODUCTION
The algorithmic aspects of Weyl algebras were first explored by Galligo [11], Takayama
[37] and others in the mid1980s. They laid out the theory of Gršobner bases in this slightly
noncommutative setting. Since then, Gršobner bases in Weyl algebras have been widely
used for practical computations in algorithmic Dmodule theory as promoted in [32]. In
the early 1990s, KandriRody and Weispfenning [17], by isolating the features of Weyl
algebras which permit Gršobner basis theory to work, extended this theory to a larger class
of noncomutative algebras, which they termed algebras of solvable type over a given
coefficient field K. This class of algebras includes the universal enveloping algebras of
