Elimination Theory for Solvable Polynomial Algebras and Their Free Modules

TL;DR

This paper develops an elimination theory for solvable polynomial algebras and their free modules using Gr"obner basis techniques, enabling systematic handling of left ideals and submodules.

Contribution

It introduces a novel elimination framework for left ideals and submodules in solvable polynomial algebras based on Gr"obner basis theory.

Findings

Established elimination theory for left ideals in solvable polynomial algebras.

Developed elimination methods for submodules of free modules over these algebras.

Extended Gr"obner basis techniques to facilitate ideal and submodule elimination.

Abstract

Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. Based on the Gr\"obner basis theory for $A$ and for free modules over $A$, an elimination theory for left ideals of $A$ and an elimination theory for submodules of free $A$-modules are established.