Groebner bases for the polynomial ring with infinite variables and their applications

TL;DR

This paper extends the theory of Gr"obner bases to polynomial rings with infinitely many variables and demonstrates their application in reconstructing partition correspondences using division algorithms.

Contribution

It introduces a framework for Gr"obner bases in infinite-variable polynomial rings and applies it to combinatorial partition problems.

Findings

Established a theory of Gr"obner bases for infinite-variable polynomial rings.

Reconstructed set correspondences among partitions using division algorithms.

Demonstrated applications in combinatorics and algebra.

Abstract

We develop the theory of Gr\"obner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division algorithm.