Partition identities and application to infinite dimensional Groebner basis and viceversa

TL;DR

This paper explores the connection between partition identities and Groebner bases of differential ideals, revealing new relationships and properties, including non-differential finiteness and identities involving colored partitions.

Contribution

It establishes a link between partition identities and Groebner basis computation for differential ideals, and provides new proofs and identities involving colored partitions.

Findings

Groebner basis of {x_1^2} relates to Rogers-Ramanujan partitions

The basis is not differentially finite under weighted lex order

Identities involving two-colored partitions are derived

Abstract

In the first part of this article, we consider a Groebner basis of the differential ideal {x_1^2} with respect to "the" weighted lexicographical monomial order and show that its computation is related with an identity involving the partitions that appear in the first Rogers-Ramanujan identity. We then prove that a Grobener basis of this ideal is not differentially finite in contrary with the case of "the" weighted reverse lexicographical order. In the second part, we give a simple and direct proof of a theorem of Nguyen Duc Tam about the Groaner basis of the differential ideal {x_1y_1}; we then obtain identities involving partitions with 2 colors.