Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristic

TL;DR

This paper explores the challenge of developing ranking functions for resolution of singularities in positive characteristic, using evolutionary algorithms to discover candidate functions and proposing conjectural solutions.

Contribution

It introduces a novel experimental approach with evolutionary search to find ranking functions in positive characteristic, leading to conjectural delayed ranking functions.

Findings

Discovered a five-component lexicographic ranking function satisfying descent criteria.

Developed conjectural delayed ranking functions for characteristic 3.

Demonstrated the effectiveness of evolutionary search in algebraic geometry problems.

Abstract

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.