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Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristic

This paper utilizes the evolutionary search model AlphaEvolve to discover a novel discretized ranking function that successfully overcomes characteristic-specific obstructions like the kangaroo phenomenon in dimension 4, leading to two conjectures regarding delayed ranking functions for resolving singularities in positive characteristic.

Original authors: Gergely Bérczi

Published 2026-02-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gergely Bérczi

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

The Big Problem: Smoothing Out Bumpy Shapes

Imagine you have a piece of crumpled paper or a craggy mountain range. In mathematics, these "bumps" and "crinkles" are called singularities. They are points where the shape behaves badly and breaks the usual rules of geometry.

Mathematicians have a tool called a blow-up to fix these bumps. Think of a blow-up like zooming in on a specific point and replacing that single crumpled spot with a whole new, smooth surface (like unfolding a crumpled piece of paper into a flat sheet).

In the world of "characteristic zero" (a specific type of math universe), mathematicians have a perfect recipe for this. They have a ranking function, which is like a scorecard. Every time they perform a blow-up, the scorecard goes down. Since the score can't go down forever (it hits zero), they know the process will eventually finish, leaving them with a perfectly smooth shape.

The Problem in "Positive Characteristic"

However, in a different math universe called positive characteristic (which behaves more like a digital clock that wraps around after hitting a certain number), this scorecard breaks.

Sometimes, when you try to smooth out a bump, the score doesn't go down. Instead, it stays the same for a long time, or it even jumps up temporarily. This is called the "Kangaroo phenomenon." It's like trying to climb down a staircase, but every few steps, a kangaroo hops you back up to a higher step. Because the score doesn't strictly go down, mathematicians have been stuck for decades, unable to prove that the smoothing process will ever finish.

The Experiment: Teaching a Robot to Find a New Scorecard

This paper is not a traditional math proof. Instead, it's a report on a computer experiment. The author, Gergely Berczi, used a powerful AI tool called AlphaEvolve to act as a "mathematical detective."

Here is how the experiment worked:

The Simulator: The author built a simplified video game. In this game, there are "bumpy shapes" (singularities) and a set of rules for how to "blow them up."
The Challenge: The AI's job was to invent a new scorecard (ranking function). This scorecard had to look at the current state of the shape and assign it a score.
The Goal: The AI needed to find a scorecard where, even if the score jumps up occasionally (like the Kangaroo), it eventually goes down enough to guarantee the process finishes. The rule was: "The score must improve at least once every 5 steps."

The Journey: Trial, Error, and Evolution

The AI didn't get it right immediately. It went through several rounds of "evolution":

Round 1 (The Broad Search): The AI tried to solve a huge variety of bumpy shapes all at once. It failed. The problem was too messy; the AI couldn't find a single rule that worked for everything.
Round 2 (Focusing In): The author told the AI to focus on a specific, tricky type of shape: 4-dimensional shapes in a specific math universe (where the numbers wrap around every 3).
The Discovery: The AI evolved a 5-part scorecard. Instead of just one number, it used a list of five numbers (like a tuple: (5, 2, 1, 0, 0)).
- The first number checks if the shape is already smooth.
- The next numbers act as tie-breakers and "plateau breakers" to handle the Kangaroo jumps.
- Crucially, this scorecard successfully guided the simulation to a smooth finish 100% of the time on a test set of 100 difficult shapes.

The "Kangaroo" and the "Antelope"

The paper uses a cute metaphor to explain the difficulty:

The Antelope: A point where the shape looks like it's getting better.
The Kangaroo: A point where, after you think you've fixed it, the math "jumps" you back to a worse state (the residual order increases).
The AI's new scorecard is smart enough to ignore these temporary jumps and look at the bigger picture, ensuring that over a short window of time, the shape is definitely getting smoother.

The Results: Conjectures, Not Proofs

The paper does not claim to have solved the 60-year-old math problem. Instead, it claims to have found strong evidence that a solution exists.

The author presents two Conjectures (educated guesses):

Conjecture 1: A specific list of five numbers (the one the AI found) works as a scorecard for a specific family of shapes.
Conjecture 2: This scorecard can be tweaked to work for even more complex shapes.

The Takeaway

Think of this paper as a proof of concept. The author didn't just sit at a chalkboard; they built a digital playground and let an AI run thousands of simulations. The AI found a "magic scorecard" that successfully navigates the tricky "Kangaroo jumps" of positive characteristic geometry.

While this isn't a final mathematical theorem yet, it provides a concrete, working example of how such a scorecard could look. It suggests that if mathematicians can formalize the AI's discovery, they might finally be able to prove that you can smooth out any shape, even in this difficult math universe.

In short: The paper uses AI to play a game of "smooth the shape" and found a winning strategy that works perfectly in a specific, difficult scenario, offering a new path forward for a major unsolved math problem.

Technical Summary: Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristic

Problem Statement
The resolution of singularities in positive characteristic remains a central open problem in algebraic geometry. While Hironaka established the existence of resolution in characteristic zero (1964) via constructive algorithms driven by ranking functions (invariants that strictly decrease along blow-up sequences), no such general ranking function is known for positive characteristic. In positive characteristic, classical invariants often fail to decrease monotonically due to Frobenius-specific pathologies, most notably the "kangaroo phenomenon," where secondary invariants (such as the residual order or "shade") can temporarily increase after a blow-up. Existing approaches struggle to circumvent these obstructions, particularly in dimensions $d \ge 4$ .

Methodology
This paper reports a series of computational experiments using AlphaEvolve, an evolutionary search engine that optimizes Python scripts within a reinforcement learning framework. The authors designed a "toy canonical blow-up simulator" to investigate whether a ranking function could be discovered that induces finite termination despite temporary increases in invariants.

The methodology involved:

Simulator Design: A deterministic simulator was constructed for hypersurface singularities in affine space $A^d$ with a distinguished elimination variable $z$ . The state consists of an ideal specification (encoded by exponent vectors and symbolic tags) and a boundary function tracking exceptional divisors. The simulator follows a fixed, deterministic rule for selecting blow-up centers based on Newton-combinatorial data.
Feature Engineering: A dictionary of 26 intrinsic numerical invariants (features $f_0$ through $f_{25}$ ) was defined as proxies for geometric quantities used in resolution theory (e.g., order, weighted order, Hilbert-Samuel values, Jacobian vanishing, and boundary data). These features were designed to be history-free (pure functions of the current state).
Evolutionary Search: AlphaEvolve was tasked with evolving a ranking function $R: \mathbb{R}^{26} \to \mathbb{R}^k$ (a tuple of values) that satisfies a bounded-delay descent criterion. Unlike the strict step-by-step descent required in characteristic zero, this criterion requires that the "best-so-far" rank strictly improves at least once within every window of $m$ steps (where $m=5$ ).
Iterative Refinement: The experiments proceeded through a curriculum. The authors manually refined the test benchmarks and feature sets after each run, adding adversarial counterexamples found by the system or external tools (like LLMs) to stress-test the evolved ranking functions.

Key Contributions and Results

Experimental Framework: The paper establishes a workflow for using evolutionary search to discover candidate mathematical constructions for birational geometry problems, specifically addressing the difficulty of finding invariants in positive characteristic.
Benchmark Construction: The authors curated a focused benchmark of 100 hypersurface singularities in dimension $d=4$ and characteristic $p=3$ . These are monic, purely inseparable singularities of the form $z^3 + \dots$ , designed to exhibit plateau behaviors and kangaroo-like jumps.
Evolved Ranking Functions:
- Continuous Ranker: An initial experiment produced a two-component continuous ranking function that achieved zero violations on a 71-instance benchmark.
- Discretized Ranker: A subsequent experiment evolved a five-component lexicographic ranking function combined with a discretization map $\Pi: \mathbb{R}^5 \to \mathbb{N}^5$ . This function achieved zero violations on the 71-instance benchmark.
- Adversarial Improvement: When a counterexample (Lemma 2) was found that violated the bounded-delay condition for the 71-instance ranker, the benchmark was expanded to 100 instances. AlphaEvolve successfully evolved a new five-component ranker ( $R_{100}$ ) that achieved zero violations on all 100 instances.
Counterexamples and Limitations: The paper explicitly documents counterexamples where the evolved rankers fail on broader classes of inputs (e.g., non-monic or higher-degree mixed terms), demonstrating that the current solutions are specific to the restricted "Weierstrass family" used in the benchmark.

Conjectures
The experimental results motivate two specific conjectures regarding the existence of delayed ranking functions:

Conjecture 1: The specific five-component lexicographic tuple evolved in the first phase ( $R_{lex}$ ) satisfies the bounded-delay descent condition for the restricted family of monic Weierstrass polynomials $z^3 + g + z^2h$ .
Conjecture 2: There exists a five-component raw rank function $R$ built from the intrinsic invariants such that its discretization $\Pi \circ R$ is a bounded-delay ranking function for the class of monic purely inseparable hypersurface singularities in dimension 4 and characteristic 3. The authors conjecture that the specific ranker $R_{100}$ may generalize to families with higher-degree mixed terms.

Significance and Claims
The paper modestly frames itself not as a finished mathematical theory, but as an exploration of how to design and evolve resolution experiments.

Proof of Concept: It demonstrates that AlphaEvolve can serve as a productive engine for mathematical exploration, generating candidate constructions that can be verified and potentially promoted to rigorous theorems.
Insight into Pathologies: The experiments highlight that in positive characteristic, termination mechanisms likely require not just order-based invariants, but a complex interplay of boundary-aware data (exceptional divisors), Jacobian signals, and p-adic depth information to break plateaus.
Methodological Shift: The work suggests a shift from enforcing strict monotonicity to accepting "bounded-delay" descent, acknowledging that geometric progress in positive characteristic may involve temporary regressions in secondary invariants.

The authors emphasize that the mathematical conclusions, conjectures, and counterexamples presented arise from independent mathematical analysis of the computational outputs, rather than being direct outputs of the AI. The paper serves as a report on a computational journey to investigate a central problem in birational geometry.