A Grothendieck ring of finite characteristic

TL;DR

This paper constructs structures in finite characteristic whose Grothendieck rings are isomorphic to polynomial rings over finite cyclic groups, demonstrating the existence of such algebraic structures with specific Grothendieck ring properties.

Contribution

It introduces a method to build structures with Grothendieck rings isomorphic to $(bZ/NbZ)[X]$ for any positive integer N, expanding understanding of Grothendieck rings in finite characteristic.

Findings

Existence of structures with Grothendieck ring $(bZ/NbZ)[X]$ for all N

Construction of bijections without cycles between sets and their complements

Demonstration of structures with non-zero Grothendieck rings in finite characteristic

Abstract

We construct, for every integer $N\in\mathbb{N}^*$, a structure whose Grothendieck ring is isomorphic to $(\mathbb{Z}/N\mathbb{Z})[X]$, thus proving the existence of structures with a non-zero Grothendieck ring with non-zero characteristic. Namely, this structure consists of the bijection without cycles between a set and a complement of $N$ points in this set.