A mathematical preprint gives a conditional route to preserving a property called strong local stability after an Artin–Schreier base-change. Its central theorem says that if every Frobenius base-change indexed from 1 onward meets the condition, then the Artin–Schreier version meets it as well. The result comes from a formal argument about algebraic objects, not an empirical sample.
The setting is positive characteristic and involves families of pairs over smooth curves. The note follows discrepancies for divisors after Artin–Schreier and Frobenius base-changes, using ramification data from those extensions and from degree-p purely inseparable extensions of discrete valuation rings. Put simply, it asks what happens to the same family when the underlying algebraic setting is replaced by one of these specified extensions.
The condition comes first
The proof is built around ramification, the behavior of these local extensions, rather than an explicit algebraic presentation of the normalized overring. It uses information recorded on the base discrete valuation ring to carry out the comparison. That makes the paper a study of formal relationships among local extensions, divisors and discrepancies, not of measured outcomes.
The proof turns on an invariant written λ(f). It identifies one specific ramification type, called ferocious, in an Artin–Schreier extension: that type occurs exactly when the characteristic parameter p divides λ(f). In the same notation, λ(f) is the ramification invariant associated with the Artin–Schreier defining function f. In the wild case, the ramification break is one plus the negative of λ(f).
Ramification sets the rules
Another local calculation concerns a degree-p purely inseparable extension of discrete valuation rings. The ramification index is p exactly when the associated invariant is prime to p. That condition is part of the note’s comparison of the two kinds of local extension.
The discrepancy formulas then split by ramification type. Under wild ramification, the discrepancy at D is p times the discrepancy at E minus (p minus 1) times (sm plus λ minus 1). Under ferocious ramification, it is the discrepancy at E minus the fraction (p minus 1) divided by p, multiplied by (sm plus λ). For a tame divisor, the formula is simpler: p times the original discrepancy plus p minus 1.
A narrow result with clear limits
One of the proof’s important reductions is eventual tameness. Every wild divisor becomes tame after sufficiently many Frobenius base-changes, with the statement applying for all sufficiently large Frobenius indices. The result is qualitative: it says after sufficiently many changes without specifying a numerical threshold.
The note also compares the Artin–Schreier side with the first Frobenius transform in a narrower setting. For a tame vertical divisor, the discrepancy after Artin–Schreier base-change equals the discrepancy of its first Frobenius transform. For a wild divisor, the stated invariance condition is equivalent to the corresponding Frobenius-side invariant being prime to p. The note gives separate formulas for γ depending on whether that invariant is prime to p or divisible by p.
A further comparison gives a lower bound only under a stated inequality between the Frobenius-side and Artin–Schreier-side invariants. When that condition holds, the resulting discrepancy is at least minus one. Because the bound depends on that inequality, it is a conditional comparison rather than a replacement for the theorem’s full assumptions.
Those qualifications define the limits of the result. The main conclusion depends on strong local stability for every Frobenius index beginning at one, and the formal argument does not establish Artin–Schreier permanence without that assumption. Nor does it test an empirical system: its objects are algebraic extensions, divisors, ramification data and discrepancies.
The document is identified as arXiv:2608.24256v2 and dated 31 Aug 2026. Its result is a conditional theorem with a ramification-based proof strategy, not a numerical finding.
Paper data and sources
Original title: Local stability of Frobenius and Artin--Schreier base-change: a comparison
Authors: Quentin Posva
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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