[Paper Review] Hilbert Functions and Low-Degree Randomness Extractors

Dave Bayer, David Mumford+3 more|arXiv (Cornell University)|Apr 11, 1993

Commutative Algebra and Its Applications46 references201 citations

TL;DR

This paper investigates the computational complexity of algebraic geometry problems, focusing on Gröbner bases and Hilbert functions as central tools. It establishes a key bound on the regularity of smooth varieties—providing a theoretical explanation for the explosive worst-case behavior of Gröbner basis computations—while advocating for deeper geometric insight to improve algorithmic efficiency in algebraic geometry and computer algebra systems.

ABSTRACT

For S ⊆ 𝔽ⁿ, consider the linear space of restrictions of degree-d polynomials to S. The Hilbert function of S, denoted h_S(d,𝔽), is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets S of arbitrary finite grids in 𝔽ⁿ with a fixed size |S|. We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size |S|. Understanding the smallest values of Hilbert functions is closely related to the study of degree-d closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree-d closures of subsets of 𝔽_qⁿ, which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022). We use the bounds on the Hilbert function and degree-d closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known.

Motivation & Objective

• To bridge the gap between theoretical computer science and algebraic geometry by aligning complexity bounds with practical computational needs.
• To explain why Gröbner basis computations exhibit explosive worst-case complexity despite being effective in practice.
• To develop computationally feasible methods for handling ideals and schemes without unnecessary field extensions or factorizations.
• To advocate for geometric intuition and resolution of singularities as essential tools in algorithmic algebraic geometry.
• To provide a foundation for understanding the complexity of core operations like ideal membership, decomposition, and normalization in algebraic geometry.

Proposed method

• Uses one-parameter group actions (λ(t)) to deform ideals via initial forms, constructing a flat family over the affine line to compute the initial ideal in(I).
• Applies Artin's criterion for flatness: checks whether syzygies of deformed generators lift properly to the central fiber.
• Employs Gröbner basis theory as the foundational computational engine for manipulating ideals and modules over polynomial rings.
• Introduces the concept of initial ideals in(I) as monomial ideals that capture scheme-theoretic limits of families under weighted monomial orders.
• Uses Serre duality and cohomology of dualizing sheaves (e.g., H^r(Ω^r_V(I))) to count absolutely irreducible components geometrically, avoiding field extensions.
• Applies the Cayley-Bacharach theorem for plane curves: V(f) is absolutely irreducible iff for every ordinary double point P, a degree d−3 curve passes through all other double points.

Experimental results

Research questions

• RQ1 Why do Gröbner basis computations exhibit such high worst-case complexity, and what geometric invariant controls this behavior?
• RQ2 How can one compute the scheme structure of the limit of a one-parameter family of varieties without introducing extraneous components?
• RQ3 What is the role of regularity in measuring the complexity of Gröbner basis computations, and how can it be bounded for smooth varieties?
• RQ4 Can geometric invariants like cohomology of dualizing sheaves be used to count absolutely irreducible components without factoring polynomials?
• RQ5 To what extent can resolution of singularities be made algorithmically effective and complexity-optimized in practice?

Key findings

• The second author establishes a bound on the regularity of any smooth variety, which explains the explosive worst-case complexity of Gröbner basis computations.
• The initial ideal in(I) is not generally generated by the initial forms of a given generating set {f₁,…,fᵣ}, necessitating careful lifting of syzygies to ensure flatness.
• Flatness of the family Y over A¹ is equivalent to the syzygies of the deformed generators g₁,…,gᵣ lifting correctly to the central fiber.
• For plane curves, the number of absolutely irreducible components is determined by the dimension of H¹(Ω¹_V(f)), which vanishes iff the curve is absolutely irreducible.
• The Cayley-Bacharach theorem provides a classical criterion: a plane curve is absolutely irreducible iff for every ordinary double point P, a degree d−3 curve passes through all other double points.
• Teitelbaum’s algorithm for resolving plane curve singularities is complexity-optimized, avoids unnecessary factorizations and field extensions, and is amenable to implementation.