A data structure for monomial ideals with applications to signature Gröbner bases

TL;DR

The paper introduces monomial divisibility diagrams (MDDs), a DAG-based data structure that maximally shares identical subtrees to compactly and efficiently represent monomial ideals. By converting traditional ideal trees into MDDs, it achieves fast membership tests and insertions, with theoretical complexity bounds and extensive empirical data on size and performance. The authors implement MDDs in AlgebraicSolving.jl and demonstrate substantial speed-ups in signature Gröbner basis computations, where monomial-ideal membership tests are a major bottleneck. The work also provides a rigorous formal framework, complexity analysis, and practical guidelines, highlighting when MDDs offer the strongest advantages, particularly for ideals arising from Gröbner bases. Overall, MDDs offer a practical, scalable alternative to generator lists and other tree-based structures for symbolic computation in polynomial rings.

Abstract

We introduce monomial divisibility diagrams (MDDs), a data structure for monomial ideals that supports insertion of new generators and fast membership tests. MDDs stem from a canonical tree representation by maximally sharing equal subtrees, yielding a directed acyclic graph. We establish basic complexity bounds for membership and insertion, and study empirically the size of MDDs. As an application, we integrate MDDs into the signature Gröbner basis implementation of the Julia package AlgebraicSolving.jl. Membership tests in monomial ideals are used to detect some reductions to zero, and the use of MDDs leads to substantial speed-ups.

Figures (6)

• Figure 1: Two trees representing the monomial ideal $\langle xyz, x^2, xy^2 \rangle$. The first level correspond the variable $z$, the second to $y$, and the last to $x$. The left tree is not an ideal tree because $\langle x^2, xy^2\rangle$ is not included in $\langle xy \rangle$. The right tree is an ideal tree. Note the extra path corresponding to the monomial $zx^2$.
• Figure 2: Insertion of a monomial $\alpha = x\cdot \alpha'$ in an ideal tree $t$ with $\operatorname{dom}(t) = \left\{ a, b, c \right\}$, with $a < b<c$. Pruning of redundant subtrees after insertion is not shown.
• Figure 3: The ideal tree of the monomial ideal generated in $\mathbb{N}^5$ by the leading monomials of the 55 elements of the Gröbner basis of five generic equations of degree 3, 3, 3, 3, and 2.
• Figure 4: The monomial divisibility diagram of the ideal tree of Figure \ref{['fig:ideal-tree']}.
• Figure 5: Size of the ideal tree and of the MDD across benchmark families. Each pair of points corresponds to a monomial ideal. The abscissa of a point is the size of the list of generators, in machine words, that is, the number of generators times the number of variables. The ordinate of the lower point of a pair is the size, in machine words, of the MDD in sequential form, that is, twice the number of edges plus the number of nodes. The ordinate of the upper point is the size of the ideal tree in the same sequential form. The shaded area indicates where the ordinate is smaller than the abscissa.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Lemma 3
• proof
• Theorem 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof