[Paper Review] Gröbner Basis Theory for Modules over Polynomial Rings over Fields with Valuation

Aritra Sen, Ambedkar Dukkipati|arXiv (Cornell University)|Apr 29, 2014

Polynomial and algebraic computation8 references3 citations

TL;DR

This paper extends Gröbner basis theory to free modules over polynomial rings over fields with valuation by incorporating coefficient valuations into the initial term definition, using an ecart function with values in ℕ to maintain well-ordering. It presents a Buchberger-like algorithm for computing Gröbner bases and Hilbert polynomials, demonstrating that initial submodules can remain constant in size under this framework, offering computational advantages over standard Gröbner bases.

ABSTRACT

A motivation to study Gröbner theory for fields with valuations comes from tropical geometry, for example, they can be used to compute tropicalization of varieties \citep{maclagan2009introduction}. The computational aspect of this theory was first studied in (Chen \& Maclagan, 2013). In this paper, we generalize this Gröbner basis theory to free modules over polynomial rings over fields with valuation. As the valuation of coefficients is also taken into account while defining the initial term, we do not necessarily get a monomial order. To overcome this problem we have to resort to other techniques like the use of ecart function where the codomain is the well-ordered set $\mathbb{N}$, and thereby give a method to calculate the Gröbner basis for submodules generated by homogeneous elements.

Motivation & Objective

• To generalize Gröbner basis theory for modules over polynomial rings over fields with valuation, where coefficient valuations influence initial terms.
• To overcome the failure of monomial orders due to valuation-based term ordering by introducing an ecart function with codomain ℕ.
• To develop a Buchberger-like algorithm for computing Gröbner bases of submodules generated by homogeneous elements.
• To enable computation of Hilbert polynomials for graded modules using the proposed algorithm.
• To extend the framework to free modules over ℤ/p^ℓℤ[x₁,…,xₙ] with analogous normal form and Gröbner basis algorithms.

Proposed method

• Define a valuation-based term order using the sum of coefficient valuation and weighted monomial degree, replacing standard monomial orders.
• Introduce an ecart function with values in ℕ to ensure well-ordering and enable Buchberger’s criterion in the absence of monomial orders.
• Construct a normal form algorithm for modules over polynomial rings with valued fields, using reduction steps guided by initial terms and ecart.
• Adapt the Buchberger algorithm by computing S-polynomials and reducing them modulo the current basis, ensuring remainder is zero for Gröbner basis certification.
• Apply the algorithm to compute Hilbert polynomials by tracking degrees and initial submodule growth in graded settings.
• Extend the framework to ℤ/p^ℓℤ[x₁,…,xₙ]-modules by adapting the normal form and Gröbner basis computation using modular arithmetic and valuation-like properties.

Experimental results

Research questions

• RQ1 Can Gröbner basis theory be generalized to modules over polynomial rings over fields with valuation, where coefficient valuations affect term ordering?
• RQ2 How can one maintain well-ordering and compute Gröbner bases when standard monomial orders fail due to valuation-based term comparison?
• RQ3 What is the computational advantage of this valuation-based Gröbner basis approach in terms of basis size and efficiency?
• RQ4 Can Hilbert polynomials be effectively computed using this new framework for graded modules?
• RQ5 To what extent can this theory be extended to finite rings like ℤ/p^ℓℤ[x₁,…,xₙ]?

Key findings

• The proposed method yields Gröbner bases that can be significantly smaller than standard Gröbner bases, with one example showing constant-sized initial submodules despite increasing degree.
• The use of an ecart function with values in ℕ ensures well-ordering and enables a Buchberger-like criterion for Gröbner basis verification.
• The algorithm correctly computes Hilbert polynomials for graded modules by tracking the growth of initial submodules under the valuation-based order.
• The framework successfully extends to free modules over ℤ/p^ℓℤ[x₁,…,xₙ], with a normal form algorithm and Buchberger-like criterion adapted to the finite ring setting.
• The remainder of S-polynomials reduces to zero if and only if the generating set is a Gröbner basis, ensuring correctness of the algorithm.
• The method allows for coefficient blow-up mitigation by computing Gröbner bases over ℤ/p^ℓℤ and lifting to ℚ, offering a practical computational pathway.