[Paper Review] Decomposition of polynomial sets into characteristic pairs

Dongming Wang, Rina Dong+1 more|arXiv (Cornell University)|Feb 28, 2017

Polynomial and algebraic computation32 references3 citations

TL;DR

This paper proposes an algorithm to decompose any finite polynomial set into finitely many characteristic pairs—each consisting of a reduced lexicographic Gröbner basis and a minimal normal triangular set—enabling exact zero decomposition of algebraic varieties. The method exploits the structural link between Ritt characteristic sets and lexicographic Gröbner bases, with the Gröbner bases computed via term order conversion and the triangular sets derived as by-products, yielding well-structured, computationally useful representations of solution sets.

ABSTRACT

A characteristic pair is a pair (G,C) of polynomial sets in which G is a reduced lexicographic Groebner basis, C is the minimal triangular set contained in G, and C is normal. In this paper, we show that any finite polynomial set P can be decomposed algorithmically into finitely many characteristic pairs with associated zero relations, which provide representations for the zero set of P in terms of those of Groebner bases and those of triangular sets. The algorithm we propose for the decomposition makes use of the inherent connection between Ritt characteristic sets and lexicographic Groebner bases and is based essentially on the structural properties and the computation of lexicographic Groebner bases. Several nice properties about the decomposition and the resulting characteristic pairs, in particular relationships between the Groebner basis and the triangular set in each pair, are established. Examples are given to illustrate the algorithm and some of the properties.

Motivation & Objective

• To develop an algorithmic method for decomposing arbitrary polynomial sets into structured representations of their zero sets.
• To establish a formal connection between Ritt characteristic sets and lexicographic Gröbner bases for improved decomposition.
• To produce characteristic pairs—(Gröbner basis, normal triangular set)—that enable exact and efficient representation of algebraic varieties.
• To ensure the resulting decomposition is finite and computationally feasible, with well-behaved algebraic properties.
• To provide a unified framework combining Gröbner basis and triangular set methods for solving polynomial systems.

Proposed method

• The algorithm computes reduced lexicographic Gröbner bases using term order conversion via the FGLM algorithm (for zero-dimensional cases) or Gröbner walk (for positive-dimensional cases).
• It leverages the inherent structural relationship between Ritt characteristic sets and lexicographic Gröbner bases to extract minimal normal triangular sets as by-products.
• The method ensures that each triangular set is normal, meaning no greatest variable appears in the leading coefficient of any polynomial in the set.
• Zero decomposition is achieved through the union of zero sets of the Gröbner bases and the zero sets of the triangular sets with respect to their initials.
• The algorithm is implemented using the FGb library for Gröbner basis computation and Maple for term order conversion and decomposition.
• Normalization of triangular sets is applied only when necessary, with efficiency improved by leveraging existing decomposition tools like the normat function.

Experimental results

Research questions

• RQ1 Can any finite polynomial set be decomposed into finitely many characteristic pairs, each combining a reduced lexicographic Gröbner basis and a minimal normal triangular set?
• RQ2 What structural relationships exist between Ritt characteristic sets and lexicographic Gröbner bases that can be exploited for decomposition?
• RQ3 How can the computation of characteristic pairs be made efficient, particularly in terms of Gröbner basis conversion and triangular set extraction?
• RQ4 What are the theoretical and computational advantages of using characteristic pairs over standard triangular or Gröbner decompositions?
• RQ5 To what extent do the resulting characteristic pairs preserve the radical ideal structure and enable exact zero decomposition?

Key findings

• The algorithm successfully decomposes any finite polynomial set into finitely many characteristic pairs, each providing a representation of the zero set via Gröbner bases and normal triangular sets.
• The computation of lexicographic Gröbner bases is the most time-consuming step, with timings ranging from 0.047s to 436.7s depending on the example and dimension.
• For zero-dimensional systems like Cyclic6 and fabrice24, the method completes in under 4 minutes, while higher-dimensional or complex systems (e.g., Wang16, filter9) exceed 4000 seconds or cause kernel loss.
• The Gröbner walk implementation in Maple is identified as a major bottleneck, significantly slowing down the overall computation.
• The resulting characteristic pairs yield exact zero decompositions: Z(P) = Z(G₁) ∪ … ∪ Z(Gₜ) = Z(C₁/inᵢ(C₁)) ∪ … ∪ Z(Cₜ/inᵢ(Cₜ)), preserving radical ideal structure.
• Normal triangular sets are obtained as by-products of Gröbner basis computation, reducing the need for separate normalization steps.