From Ideal Membership Problem for polynomial rings to Dehn Functions of Metabelian Groups

TL;DR

This paper explores the complexity of the ideal membership problem in Laurent polynomial rings and links it to the Dehn functions of metabelian groups, aiming to understand their computational and geometric properties.

Contribution

It introduces a complexity function for the ideal membership problem and connects it to Dehn functions, proposing a method to construct metabelian groups with superexponential Dehn functions.

Findings

Defined a complexity function for ideal membership

Connected this complexity to Dehn functions of metabelian groups

Aimed to construct groups with superexponential Dehn functions

Abstract

The ideal membership problem asks whether an element in the ring belongs to the given ideal. In this paper, we propose a function that reflecting the complexity of the ideal membership problem in the ring of Laurent polynomials with integer coefficients. We also connect the complexity function we define to the Dehn function of a metabelian group, in the hope of constructing a metabelian group with superexponential Dehn function.