Discretized ranking-function conjecture for purely inseparable hypersurface singularities

Let Π\Pi be the discretization map from the paper, and let f0,,f25f_0,\ldots,f_{25} be the intrinsic invariants used to construct a five-component raw rank function R=(c1,c2,c3,c4,c5)R=(c_1,c_2,c_3,c_4,c_5). A bounded-delay ranking function is a discretized rank whose ranking improvements occur within a uniformly bounded number of steps. Discretized ranking-function conjecture. There exists such an RR, built from f0,,f25f_0,\ldots,f_{25}, for dimension-44, characteristic-33 monic purely inseparable hypersurface singularities, with

ΠR\Pi\circ R

a bounded-delay ranking function. Moreover, the explicit ranker R100\mathcal R_{100} generalizes to families obtained by adding further mixed terms of arbitrarily high degree. This conjecture proposes a uniform ranking mechanism for the canonical blow-up process, but the supplied text gives only computational evidence and does not establish either the existence claim or the asserted generalization.

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Primary source

Gergely Bérczi, “Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristic”, arXiv:2602.06553 (2026).

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