Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristic

Resolution of singularities in positive characteristic remains a long-standing open problem in algebraic geometry. In characteristic zero, the problem was solved by Hironaka in 1964, work for which he was awarded the Fields Medal. Modern proofs proceed by constructing suitable ranking functions, that is, invariants shown to strictly decrease along canonical sequences of blow-ups, ensuring termination. In positive characteristic, however, no such general ranking function is known: Frobenius-specific pathologies, such as the kangaroo phenomenon, can cause classical characteristic-zero invariants to plateau or even temporarily increase, presenting a fundamental obstruction to existing approaches. In this paper we report a sequence of experiments using the evolutionary search model AlphaEvolve, designed to discover candidate ranking functions for a toy canonical blow-up process. Our test benchmarks consist of carefully selected hypersurface singularities in dimension $4$ and characteristic $p=3$, with monic purely inseparable leading term, a regime in which naive order-based invariants often fail. After iteratively refining the experimental design, we obtained a discretized five-component lexicographic ranking function satisfying a bounded-delay descent criterion with zero violations across the benchmark. These experiments in turn motivated our main results: the conjectural delayed ranking functions in characteristic $3$ formulated in two conjectures.

💡 Research Summary

The paper tackles one of the most stubborn open problems in algebraic geometry: constructing a canonical resolution of singularities in positive characteristic. While Hironaka’s 1964 theorem settled the characteristic‑zero case and subsequent work (Bierstone‑Milman, Villamayor, Włodarczyk, etc.) produced well‑behaved ranking (or invariant) functions that strictly decrease along a prescribed sequence of blow‑ups, the same approach breaks down in characteristic p > 0. Frobenius phenomena, most famously the “kangaroo” effect, cause classical invariants such as order or residual order to plateau or even increase temporarily, preventing a straightforward well‑founded descent argument.

To explore whether a suitable invariant can be discovered automatically, the author employs AlphaEvolve, an evolutionary search engine that treats Python scripts as individuals, mutates a designated “evolve block” using large language models, and evaluates each candidate with a user‑provided scoring harness. The harness simulates a toy canonical blow‑up process on a fixed benchmark of hypersurface singularities, selects blow‑up centers deterministically, and computes a fixed set of 26 numerical features (orders, coefficient‑ideal data, Hilbert‑Samuel multiplicities, Frobenius‑related quantities, etc.). The ranking function to be evolved is a pure function from this feature vector to a tuple that is compared lexicographically; it must be memory‑less and deterministic.

The experimental pipeline proceeds in several stages:

1. Benchmark construction – Initially 71 four‑dimensional purely inseparable hypersurfaces in characteristic 3 (leading term z³) are curated to exhibit the known pathologies (kangaroo jumps, tie‑breaking instability). After discovering counter‑examples, the set is enlarged to 100 adversarial instances.

2. Feature selection – Inspired by characteristic‑zero theory and positive‑characteristic literature, 26 invariants are implemented, some directly, others via proxies that approximate costly algebraic computations.

3. AlphaEvolve runs – The scoring function rewards “bounded‑delay descent”: a candidate ranking function may allow a finite number of steps where the lexicographic tuple does not strictly decrease, but after a bounded delay it must drop. Zero violations on the benchmark yield the highest score.

4. Resulting ranking functions – The first successful model is a continuous two‑component ranker that passes the benchmark. However, because it yields non‑integer values, termination cannot be guaranteed. The author then discretizes the approach, producing a five‑component integer lexicographic ranker. Its components are roughly: (i) primary order, (ii) residual order, (iii) Frobenius exponent, (iv) height of the exceptional divisor “ladder”, and (v) a corrective term. This discretized ranker achieves zero violations on all 71 instances.

5. Refinement after a counterexample – A structural conjecture (Theorem 2 in the paper) about the discretized ranker is falsified by a concrete example. The benchmark is expanded, and a new five‑component integer ranker is evolved, again with zero violations on the full set of 100 instances.

The empirical success leads to two conjectures:

• Conjecture 1 – For any purely inseparable hypersurface of dimension ≥ 4 in any positive characteristic, there exists a finite‑dimensional integer‑valued lexicographic ranking function that satisfies a bounded‑delay descent property along the canonical blow‑up sequence.

• Conjecture 2 – Incorporating the height of the exceptional divisor and a Frobenius‑induced residual order into the ranking function suffices to guarantee bounded‑delay descent for the whole class of positive‑characteristic singularities considered.

Beyond the specific conjectures, the paper emphasizes methodological lessons. The separation of concerns—fixed simulator and feature set versus mutable ranking function—prevents “cheating” (e.g., using the number of exceptional divisors as a hidden state). The experiments also reveal that the most time‑consuming work is not tuning the evolutionary algorithm but iteratively refining the benchmark, feature set, and scoring penalties to align the optimization objective with the mathematical goal of strict descent after geometric drops.

Finally, the author reflects on the role of AI in mathematical discovery. AlphaEvolve can generate candidate constructions rapidly, but rigorous verification still rests with the mathematician. Future work could make the center‑selection rule itself learnable, potentially uncovering novel blow‑up strategies. Nonetheless, the present study demonstrates that automated search can produce concrete, verifiable invariants in a domain where traditional theory has long been stalled, offering a promising new direction for the resolution of singularities in positive characteristic.