Degree Bounds for Gr\"obner Bases in Algebras of Solvable Type

TL;DR

This paper establishes doubly-exponential degree bounds for Gr"obner bases in various algebras of solvable type, extending classical results to noncommutative settings like Weyl and universal enveloping algebras.

Contribution

It provides the first doubly-exponential degree bounds for Gr"obner bases in algebras of solvable type, generalizing known bounds from commutative and specific noncommutative cases.

Findings

Doubly-exponential degree bounds for Gr"obner bases in algebras of solvable type.

Extension of classical bounds to Weyl and universal enveloping algebras.

Application of generalized Stanley decompositions for bound computation.

Abstract

We establish doubly-exponential degree bounds for Gr\"obner 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 algebras, and universal enveloping algebras of finite-dimensional Lie algebras. For the computation of these bounds, we adapt a method due to Dub\'e 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).