Electrical networks and the Grove algebra

TL;DR

This paper introduces the grove algebra, a novel algebraic structure modeling electrical networks, and develops its combinatorial and algebraic properties, including a quadratic Gr"obner basis for its ideal.

Contribution

It defines the grove algebra as an electrical analogue of the Plücker ring and analyzes its structure using double groves and Gr"obner basis techniques.

Findings

Established the grove algebra as a ring of regular functions on electrical networks.

Derived a quadratic Gr"obner basis for the grove ideal.

Connected the algebraic structure to combinatorics of double groves.

Abstract

We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analogue of the Pl\"ucker ring studied classically in invariant theory. We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gr\"obner basis for the grove ideal.