[Paper Review] Effective Methods for the Computation of Bernstein-Sato polynomials for Hypersurfaces and Affine Varieties

Daniel de Andrés, Viktor Levandovskyy+1 more|arXiv (Cornell University)|Feb 18, 2010

Polynomial and algebraic computation22 references3 citations

TL;DR

This paper presents novel algorithms for computing Bernstein-Sato polynomials of hypersurfaces and affine varieties using a general method for intersecting left ideals with subalgebras generated by a single element. The approach, implemented in Singular, improves efficiency by replacing Gröbner basis elimination with linear dependence detection, and introduces a new method for computing the Bernstein-Sato polynomial of a variety, validated through examples including singularities and monomial curves.

ABSTRACT

This paper is the widely extended version of the publication, appeared in Proceedings of ISSAC'2009 conference \citep*{ALM09}. We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for computing an intersection of a left ideal of an associative algebra over a field with a subalgebra, generated by a single element. We show applications of this algorithm in different algebraic situations and describe our implementation in extsc{Singular}. Among other, we use this algorithm in computational $D$-module theory for computing e. g. the Bernstein-Sato polynomial of a single polynomial with several approaches. We also present a new method, having no analogues yet, for the computation of the Bernstein-Sato polynomial of an affine variety. Also, we provide a new proof of the algorithm by Briançon-Maisonobe for the computation of the $s$-parametric annihilator of a polynomial. Moreover, we present new methods for the latter computation as well as optimized algorithms for the computation of Bernstein-Sato polynomial in various settings.

Motivation & Objective

• To develop efficient computational methods for Bernstein-Sato polynomials of hypersurfaces and affine varieties in D-module theory.
• To address the computational challenge of eliminating variables in non-commutative algebras, particularly in the context of D-modules.
• To generalize the principal intersection technique to subalgebras generated by a single element for use in D-module computations.
• To provide a new, correct, and efficient algorithm for computing the Bernstein-Sato polynomial of an affine variety, with no prior analogues.
• To compare and optimize different approaches—annihilator-based, initial-based, and syzygy-driven—for computing Bernstein-Sato polynomials, especially in challenging singular cases.

Proposed method

• Proposes a general algorithm for computing the intersection of a left ideal in a non-commutative algebra with a subalgebra generated by a single element.
• Applies the principal intersection method to replace computationally expensive Gröbner basis elimination with detecting K-linear dependence of normal forms.
• Uses this technique to compute the s-parametric annihilator of a polynomial f, a key step in Bernstein-Sato polynomial computation.
• Introduces a new algorithm for computing the Bernstein-Sato polynomial of an affine variety by extending the framework of Budur et al. (2006).
• Implements the method in the computer algebra system Singular, using the Plural library for non-commutative Gröbner bases.
• Employs monomial orderings such as (dp(6), rp) and the notation x ≫ y to manage elimination in non-commutative settings.

Experimental results

Research questions

• RQ1 Can the principal intersection method be generalized to efficiently compute intersections of left ideals with subalgebras generated by a single element in non-commutative algebras?
• RQ2 How can the computation of the Bernstein-Sato polynomial of an affine variety be achieved without relying on existing methods for hypersurfaces?
• RQ3 What are the relative performance advantages of annihilator-based versus initial-based methods for computing Bernstein-Sato polynomials across different classes of singularities?
• RQ4 Can the syzygy-driven method for computing Ann_{D[s]}(f^s) be proven superior to the Briançon-Maisonobe algorithm, and under what conditions?
• RQ5 To what extent can the new method for affine varieties be generalized to more complex singularities, such as cyclic quotient singularities of higher codimension?

Key findings

• The principal intersection method significantly improves performance by replacing Gröbner basis elimination with linear algebra over K, reducing computational complexity.
• The new algorithm for computing the Bernstein-Sato polynomial of an affine variety is correct and provides the first method without analogues in the literature.
• For the monomial curve Spec ℂ[T³,T⁴,T⁵], the computed b_Z(s) matches Shibuta (2008): (s+1)²(s+7/9)(s+5/9)(s+1/2)(s+8/9)(s+11/9)(s+10/9)(s+4/9).
• For the Hirzebruch-Jung singularity of type (5,2), the Bernstein-Sato polynomial is b_Z(s) = (s+1)³(s+4/3)(s+5/3)(s+3/2), confirming known results.
• The method fails to compute the Bernstein-Sato polynomial for the cyclic quotient singularity of type (6;1,2,3) with 9 generators, indicating computational limits in high-dimensional or complex cases.
• The annihilator-based method outperforms the initial-based method for non-hyperplane arrangements, while the reverse holds for hyperplane arrangements, suggesting context-dependent efficiency.