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DEGREE BOUNDS FOR GR OBNER BASES IN ALGEBRAS OF SOLVABLE MATTHIAS ASCHENBRENNER AND ANTON LEYKIN
 

Summary: DEGREE BOUNDS FOR GR šOBNER BASES IN ALGEBRAS OF SOLVABLE
TYPE
MATTHIAS ASCHENBRENNER AND ANTON LEYKIN
ABSTRACT. We establish doubly-exponential degree bounds for Gršobner bases in certain
algebras of solvable type over a field (as introduced by Kandri-Rody and Weispfenning).
The class of algebras considered here includes commutative polynomial rings, Weyl alge-
bras, and universal enveloping algebras of finite-dimensional 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 doubly-exponential 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 mid-1980s. They laid out the theory of Gršobner bases in this slightly
non-commutative setting. Since then, Gršobner bases in Weyl algebras have been widely
used for practical computations in algorithmic D-module theory as promoted in [32]. In
the early 1990s, Kandri-Rody 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 non-comutative algebras, which they termed algebras of solvable type over a given
coefficient field K. This class of algebras includes the universal enveloping algebras of

  

Source: Aschenbrenner, Matthias - Department of Mathematics, University of California at Los Angeles

 

Collections: Mathematics