[Paper Review] Binomial Ideals

David Eisenbud, Bernd Sturmfels|arXiv (Cornell University)|Jan 11, 1994

Commutative Algebra and Its Applications282 citations

TL;DR

This paper investigates binomial ideals—polynomial ideals generated by binomials—and establishes that their radicals, associated primes, and isolated primary components are also binomial. It proves that binomial ideals admit primary decompositions using binomial primary ideals, providing a geometric characterization of affine varieties defined by binomials and enabling sparsity-preserving algorithms for radical and primary decomposition.

ABSTRACT

We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and it has numerous applications within and beyond pure mathematics. The ideals defining toric varieties are precisely the binomial prime ideals. Our main results concern primary decomposition: If $I$ is a binomial ideal then the radical, associated primes, and isolated primary components of $I$ are again binomial, and $I$ admits primary decompositions in terms of binomial primary ideals. A geometric characterization is given for the affine algebraic sets that can be defined by binomials. Our structural results yield sparsity-preserving algorithms for finding the radical and primary decomposition of a binomial ideal.

Motivation & Objective

• To understand the algebraic and geometric structure of ideals generated by binomials.
• To determine whether the radicals, associated primes, and primary components of binomial ideals remain binomial.
• To provide a geometric characterization of affine algebraic sets definable by binomial ideals.
• To develop efficient, sparsity-preserving algorithms for computing the radical and primary decomposition of binomial ideals.
• To extend the theory of binomial ideals with applications in algebraic geometry and beyond.

Proposed method

• Analyzing the algebraic structure of binomial ideals using commutative algebra techniques, particularly primary decomposition theory.
• Proving that the radical and associated primes of a binomial ideal are themselves binomial ideals, using properties of monomial and binomial generators.
• Establishing that isolated primary components of binomial ideals are also binomial, relying on the structure of prime ideals in polynomial rings.
• Constructing a primary decomposition of a binomial ideal using binomial primary ideals, preserving the sparsity of the original generators.
• Characterizing affine algebraic sets defined by binomials via geometric and combinatorial conditions on their defining ideals.
• Designing algorithms that exploit the binomial structure to compute the radical and primary decomposition efficiently, maintaining sparsity in intermediate steps.

Experimental results

Research questions

• RQ1 Are the radicals and associated primes of a binomial ideal necessarily binomial?
• RQ2 Can a binomial ideal be decomposed into primary components that are themselves binomial ideals?
• RQ3 What geometric conditions characterize affine algebraic sets defined by binomial ideals?
• RQ4 How can the radical and primary decomposition of a binomial ideal be computed while preserving sparsity?
• RQ5 What structural properties of binomial ideals allow for algorithmic efficiency in decomposition tasks?

Key findings

• The radical of a binomial ideal is a binomial ideal, preserving the binomial structure under radical operations.
• The associated primes of a binomial ideal are also binomial ideals, indicating a strong closure property under prime decomposition.
• Isolated primary components of a binomial ideal are binomial, confirming that binomiality is preserved in key decomposition components.
• Every binomial ideal admits a primary decomposition into binomial primary ideals, establishing a fundamental structural result.
• Affine algebraic sets defined by binomial ideals are characterized by specific geometric and combinatorial conditions on their defining ideals.
• Sparsity-preserving algorithms for computing the radical and primary decomposition of binomial ideals are achievable, leveraging the binomial structure to reduce computational complexity.