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An algorithm for annihilator and Bernstein-Sato polynomial of a rational function

This paper presents an algorithm implemented in SINGULAR that computes the Bernstein-Sato polynomial of a rational function by determining the annihilator of its numerator-denominator pair, thereby generating explicit non-trivial examples and supporting existing conjectures.

Original authors: Manuel González-Villa, Edwin León-Cardenal, Viktor Levandovskyy, Jorge Martín-Morales

Published 2026-02-04

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CC BY 4.0

Original authors: Manuel González-Villa, Edwin León-Cardenal, Viktor Levandovskyy, Jorge Martín-Morales

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery involving a very specific kind of mathematical object: a rational function. In simple terms, this is just a fraction where both the top (numerator) and bottom (denominator) are polynomials (expressions like $x^2 + 3x + 1$ ).

For decades, mathematicians have studied the "DNA" of these fractions to understand their hidden shapes and singularities (points where things break or get messy). A key part of this DNA is something called the Bernstein-Sato polynomial. Think of this polynomial as a unique "fingerprint" or a "secret code" that reveals deep secrets about the function's behavior.

The Problem: A Missing Key

Until now, while mathematicians knew this fingerprint existed for simple fractions, they didn't have a reliable way to actually calculate it for complex rational functions. They only knew how to do it for very simple, "trivial" cases. It was like knowing a treasure map existed but having no compass to find the treasure.

The Solution: A New Algorithm

The authors of this paper, a team of mathematicians, have built a new algorithm (a step-by-step recipe for a computer) to find this fingerprint. Here is how they did it, explained through analogies:

1. The "Shadow" Strategy (The Annihilator)

To find the fingerprint of the fraction $\frac{f}{g}$ , the authors realized they couldn't just look at the fraction directly. Instead, they looked at the "shadow" cast by the numerator ( $f$ ) and the denominator ( $g$ ) separately.

The Analogy: Imagine you want to understand a complex machine made of two gears. Instead of trying to take the whole machine apart, you study how each gear spins on its own.
The Math: They first calculated the "annihilator" of the pair $(f, g)$ . In math-speak, an annihilator is a set of rules (differential operators) that, when applied to the function, make it vanish (turn to zero). They found the rules for the pair, and then tried to adapt those rules to work for the fraction.

2. The "Squeezing" Problem (Saturation)

When they tried to adapt the rules from the separate gears to the whole fraction, they hit a snag. The rules they got were "loose" or "incomplete." They were missing some crucial constraints.

The Analogy: Imagine you have a net to catch fish. The net you built from the separate gears has holes in it. You need to "squeeze" the net tighter to catch the specific fish you are looking for.
The Math: This process is called saturation. The authors developed a method to "tighten" the net. They proved that if a certain condition is met (which they call the " $C_m$ -condition"), you can simply tighten the net to get the perfect set of rules.

3. The "Backup Plan" (When the Condition Fails)

What if the condition isn't met? What if the net is too broken to just tighten?

The Analogy: If the net is too torn, you don't give up. You use a different tool: a "sieve" that filters out the bad parts layer by layer.
The Math: The authors created a fallback method. If the simple "tightening" doesn't work, they use a recursive process (a loop that repeats itself) to peel away the extra noise and find the true annihilator. This ensures the algorithm works even in the trickiest cases.

The Result: Cracking the Code

Once they have the perfect set of rules (the annihilator), finding the fingerprint (the Bernstein-Sato polynomial) becomes a matter of solving a linear algebra puzzle.

The Analogy: Once you have the perfect net, you just throw it in the water and see what pattern of fish it catches. That pattern is your fingerprint.
The Outcome: The team implemented this entire process in a free computer software called Singular. They tested it on several examples that were previously impossible to solve.
- They found that for some fractions, the fingerprint is surprisingly simple.
- For others, it requires a long chain of steps (many differential operators) to reveal the answer.
- They confirmed that the roots of these fingerprints (the numbers that make the polynomial zero) relate to the "monodromy" of the function—a concept related to how the function twists and turns around its singular points.

Summary

In short, this paper provides the instruction manual for a computer to automatically discover the hidden "fingerprint" of any rational function. Before this, mathematicians were stuck with only a few simple examples. Now, they have a powerful, automated tool that can handle complex fractions, verify existing theories, and uncover new mathematical patterns that were previously invisible.

Technical Summary: An Algorithm for Annihilator and Bernstein-Sato Polynomial of a Rational Function

Problem Statement
The paper addresses the computational challenge of determining the Bernstein-Sato polynomial for rational functions, specifically the quotient $f/g$ of two polynomials. While the theory of $b$ -functions (Bernstein-Sato polynomials) for single polynomials is well-established, with algorithms existing for their computation, the extension to rational functions is a more recent development introduced by Takeuchi [33]. Prior to this work, only trivial examples of Bernstein-Sato polynomials for rational functions were known. The core difficulty lies in computing the annihilator of the module generated by $(f/g)^s$ within the ring of differential operators with polynomial coefficients, $D[s]$ , and subsequently deriving the functional equation that defines the polynomial.

Methodology
The authors propose a strategy that reduces the problem of finding the annihilator of a rational function to the study of the annihilator of a pair of polynomials $(f, g)$ . The methodology proceeds through the following steps:

Annihilator of the Pair: The algorithm first computes the annihilator ideal $I(s_1, s_2) = \text{Ann}_{D[s_1, s_2]}(f^{s_1}g^{s_2})$ in the ring of differential operators with two auxiliary variables.
Substitution and Saturation: By substituting $s_1 = s$ and $s_2 = -s - m$ (where $m$ is a non-negative integer related to the denominator's power), the authors obtain an ideal $I(s, -s-m)$ in $D[s]$ . However, this ideal is generally not $C[s]$ -saturated and thus does not directly equal the desired annihilator $\text{Ann}_{D[s]}(g^{-m}(f/g)^s)$ .
Central Saturation: The paper utilizes an algorithm for computing the central saturation of a left ideal with respect to the commutative subring $C[s]$ . This involves computing a Gröbner basis and performing a saturation operation to recover the full annihilator.
Handling Non-Vanishing Conditions: A critical theoretical contribution is the identification of a "non-vanishing condition" on the Bernstein-Sato ideal of the pair $(f, g)$ $(f, g)$ . Specifically, the authors define a $C_m$ $C_{m}$ -condition related to the factors of the Bernstein-Sato ideal $B_{f,g}$ $B_{f, g}$ .
- If $B_{f,g}$ satisfies the $C_m$ -condition, the annihilator of the rational function is exactly the saturation of $I(s, -s-m)$ .
- If the condition fails, the authors provide a recursive method (Lemma 4.9) to compute the annihilator by relating it to annihilators of higher powers of the denominator, effectively stepping through the failure of the condition.
Bernstein-Sato Polynomial Computation: Once the annihilator is determined, the Bernstein-Sato polynomial $b_{f/g, m}^{(N)}(s)$ is computed by finding the minimal polynomial relation in the quotient module defined by the annihilator and the ideal generated by terms involving $f^k g^{N-k}$ . This is implemented via linear algebra on the normal forms of powers of $s$ modulo the annihilator.

Key Contributions and Results

Algorithmic Framework: The paper presents a complete algorithm (Algorithm 2) for computing the annihilator of $g^{-m}(f/g)^s$ and a subsequent algorithm (Algorithm 3) for computing the Bernstein-Sato polynomial. These algorithms rely on Gröbner bases in noncommutative PBW algebras.
Theoretical Conditions: Theorem 4.5 establishes a sufficient condition (the $C_m$ -condition) under which the annihilator of the rational function is simply the saturation of the substituted pair annihilator. The paper also details how to proceed when this condition is not met.
Implementation: The algorithms have been implemented in the computer algebra system Singular.
Explicit Examples: The authors provide explicit, non-trivial computations of Bernstein-Sato polynomials for specific rational functions (e.g., $f = x^2+y^3, g=xy$ $f = x^{2} + y^{3}, g = x y$ and $f = x^6+y^6+2zx^3y^3, g=z^2$ $f = x^{6} + y^{6} + 2 z x^{3} y^{3}, g = z^{2}$ ).
- In Example 6.1, the authors demonstrate cases where a large number of differential operators ( $N$ ) are required to stabilize the Bernstein-Sato polynomial.
- In Example 6.2, the computed polynomial is compared with the eigenvalues of the local monodromy, supporting the relationship between the roots of the $b$ -function and the monodromy of the rational function.
- The examples reveal that, unlike the classical case for polynomials, the roots of the Bernstein-Sato polynomial for rational functions are not necessarily negative rational numbers.

Significance
The paper claims significance primarily in providing the first general computational method for the Bernstein-Sato polynomial of rational functions, a topic that previously lacked non-trivial examples. By leveraging the relationship between the rational function and the pair of its numerator and denominator, the authors bridge the gap between existing algorithms for polynomial pairs and the more complex rational case. The work supports existing conjectures regarding the relationship between the roots of the Bernstein-Sato polynomial and the monodromy of the rational function. Furthermore, the implementation in Singular allows for the exploration of these invariants in concrete cases, facilitating further research into the singularity theory of rational functions. The authors remain modest, noting that while they support existing conjectures, the full scope of the theory (such as the behavior of roots for all $m$ ) remains an area for further investigation.