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.

• The paper introduces a new explicit family of height‑2 prime ideals in k[[x, y, z]] that are minimally generated by n elements for any n ≥ 3.
• It employs combinatorial numerical semigroup analysis and binomial determinant evaluations to determine the generating polynomials without relying on Gröbner bases.
• The construction provides efficient computational methods with fully explicit formulas, enabling deeper exploration of generation properties in three-dimensional rings.

Explicit Minimal Generating Sets for Prime Ideals with Unbounded Number of Generators in 3D Power Series Rings

Introduction and Problem Context

This paper constructs a new explicit family of height-2 prime ideals in the three-dimensional formal power series ring $R = k[[x, y, z]]$ over a field $k$ of characteristic zero, each minimally generated by $n$ elements for arbitrary $n \geq 3$. This extends and systematizes phenomena first observed by T.T. Moh and others regarding the existence of prime ideals in $R$ with unbounded minimal number of generators. The description of generators is made fully explicit (up to solutions of certain binomial linear systems), and is given without recourse to Gröbner bases. Key advances over previous constructions, e.g., Moh's and Maurer's, include coprimality of the exponents in the defining quasi-monomial homomorphism, a detailed combinatorial analysis leveraging the structure of numerical semigroups, and the explicit solution of the required linear systems needed to concretely determine the generating polynomials.

Main Results and Methods

Let $n \geq 3$, $a = (n-1)n/2$, and $q = 2 \lfloor (n+1)/2 \rfloor -1$. Define a $k$-algebra homomorphism $p: k[[x, y, z]] \to k[[t]]$ by $k$0, $k$1, $k$2, and let $k$3. The main theorems established are:

• $k$4 is prime of height two, with $k$5 (i.e., $k$6 is minimally generated by $k$7 elements).
• There is an explicit construction for a set $k$8 of minimal generators, where each $k$9 is determined up to solutions of at most four explicit consistent linear systems with binomial coefficients of size at most $n$0 each.
• The explicit form of $n$1 is given in terms of their $n$2-leading monomials and correction ("tail") terms, referencing the combinatorics of the semigroup $n$3.
• The leading forms of the generator polynomials are described by binomial determinants, for which explicit closed-form evaluation is available, and the combinatorial structure is optimized to guarantee uniqueness (modulo lower order terms).
• No Gröbner basis computations are required at any stage.

Combinatorial and Number-Theoretical Foundations

A fundamental aspect is the analysis of the semigroup $n$4, its factorization invariants, and related binomial determinants. The coprimality of the exponents $n$5 ensures that the image has the correct dimension and that generation properties in $n$6 lift to the polynomial ring $n$7 where needed.

The authors synthesize machinery from the theory of numerical semigroups (combinatorial stratification, factorization length, Frobenius number, etc.) to systematically identify the orders and supports of the required polynomial lead terms. This structure dictates both the degrees and the binomial combinatorics behind the construction of the generators.

Determination of Minimal Generating Sets

For each $n$8, the construction of the generators proceeds through a hierarchy:

1. Identify parameters $n$9, $n \geq 3$0, and the corresponding numerical semigroup $n \geq 3$1.
2. For a specific grading and certain intervals in $n \geq 3$2, construct $n \geq 3$3-homogeneous polynomials $n \geq 3$4 with minimal support.
3. For each $n \geq 3$5, compute correction ("tail") terms so that $n \geq 3$6 satisfies $n \geq 3$7. This is done by solving up to four systems of linear equations with binomial coefficients.
4. The explicit form of the $n \geq 3$8 is determined by binomial determinants and their combinatorics.
5. The proof that these $n \geq 3$9 generate $R$0 minimally uses length-counts, lifting and specialization arguments, explicit calculation in associated graded pieces, and Nakayama’s lemma.

Special treatment is given to edge and degenerate cases (e.g., small $R$1) with exact calculation.

Novel Computability Aspects

The authors provide a complete Python implementation (see GitHub repository), which outputs the explicit generating set for arbitrary $R$2, leveraging efficient combinatorial routines for the relevant binomial determinantal computations and linear system solving.

Structural and Theoretical Implications

• The existence of such explicit families answers longstanding questions arising from Macaulay’s, Abhyankar’s, and Moh’s foundational work on non-complete intersection primes in dimension three, extending their conclusions to explicitly presented (not just existentially) constructed cases.
• The construction demonstrates that, for any $R$3, there exists a height two prime ideal in $R$4 (and $R$5) with minimal number of generators exactly $R$6. The result is concrete and algorithmic.
• The main techniques clarify and partially resolve conjectures concerning the relationship between ring dimension and possible bounds on minimal numbers of generators for primes, contributing evidence relevant to the open Sally and Shimoda questions.
• The substitution of complicated iterative processes (as in previous literature) by direct calculation of explicit generating sets constitutes a strong claim for the tractability of generation in the context of numerical semigroup codified presentations.

Numerical and Algorithmic Advantages

• The explicitness of the minimal generating sets and the avoidance of Gröbner basis computations lead to considerable improvements in computational efficiency; it becomes feasible to compute generating sets for large $R$7, where previous approaches stall due to computational complexity.
• The coprimality of the defining exponents substantially simplifies the determination of orders of vanishing and the structure of the image of $R$8, streamlining both combinatorial and homological analysis.

Limitations and Future Directions

Several substantial theoretical and computational questions are raised:

• Extending the analysis to include explicit descriptions of the corresponding Hilbert-Burch matrices and higher syzygies.
• Determining whether analogous families exist in positive characteristic fields and whether characteristic can sometimes force the number of minimal generators to drop (see earlier work by the authors on characteristic dependence).
• Establishing if these non-complete intersection prime ideals are set-theoretic complete intersections.
• Generalizing the approach to less regular local rings or more general classes of power series/polynomial rings.

Conclusion

The authors provide a completely explicit description (and algorithm) for minimally generating a prime ideal with $R$9 generators in $n \geq 3$0 (or $n \geq 3$1) for any $n \geq 3$2, using combinatorial semigroup techniques and binomial determinant theory, and solving explicit small linear systems. The results represent strong progress in understanding generation in dimension three, with implications for both computational algebra and the theory of prime ideals in Noetherian rings. The approach offers a new paradigm for investigating questions at the intersection of commutative algebra, combinatorics, and computational methods.

See the paper for a comprehensive list; key background includes Moh [15,16], Herzog [10], Sally [19], García-Sánchez et al. [5], and computational references for SINGULAR and binomial determinants.

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Source:

"Explicit minimal generating sets of a family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring" (2604.00638)