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

Laura González, Francesc Planas-Vilanova

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 "Gröbner-free" approach.

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

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 "Gröbner-free" approach.

Paper Structure

This paper contains 7 sections, 34 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result and overview of the paper
  3. Definition and inclusion on the good kernels
  4. Ensuring when the good kernels are zero
  5. Bases for the good kernels
  6. Proof of the main result: a minimal generating set of the kernel
  7. Example and a Python code

Key Result

Theorem 1

gp1 Extending to Basis Method. Let $s=\min\{\hbox{${\rm ord}_{\sigma}$}(f)\mid f\in \ker(\rho), f\neq 0\}$, $s\geq 1$. Let $\hbox{$\mathscr{C}$}$ be a minimal generating set of $\ker(\rho)$. Let $\hbox{$\mathscr{D}$}_0,\ldots,\hbox{$\mathscr{D}$}_{a-1}\subset \ker(\rho)$ be such that, for each $i=0,

Theorems & Definitions (81)

  • Definition 2.3
  • Theorem
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Example 3.3
  • Lemma 3.5
  • proof
  • Definition 3.6
  • ...and 71 more