Explicit minimal generating sets of a family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring

TL;DR

This paper introduces a new family of prime ideals in a three-dimensional power series ring with unbounded minimal generators, providing explicit descriptions and a Python code for their analysis.

Contribution

It presents an explicit construction of prime ideals with unbounded minimal generators and a simple, Gr"obner-free method to determine their minimal generating sets.

Findings

Explicit family of prime ideals with unbounded minimal generators

Description of minimal generating polynomial sets

Python code for solving related linear systems

Abstract

We display a new family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring over a field of characteristic zero. These primes are obtained as the kernel of a quasi-monomial algebra homomorphism. Up to constant coefficients, determined by some specific linear systems with binomial entries, we describe their minimal generating polynomial sets. The advantage of our family with respect to some previous work is, on the one hand, the explicit description of the generating sets and, on the other hand, the simplicity of the exponents of the aforementioned quasi-monomial homomorphism. We also provide a code in Python which states and solves the linear systems that lead to a complete description of the minimal generating sets with a "Gr\"obner-free" approach.