Normalized Maps of Arbitrary Arity to Wound Unipotent Groups

For every integer n>=2, we construct a field k of characteristic two, a single closed point x of the projective line, and a nonconstant morphism from n copies of its complement to a smooth geometrically connected commutative k-wound unipotent group of dimension 2^n-1. The morphism is zero on every coordinate slice through infinity, while the residue field L always satisfies [L:kL^2]=2. The proof uses a determinant and the parameter-parity decomposition of a product of additive polynomials. For n>=3, this disproves the primitivity-bound assertion in Conjecture 5 of Oberwolfach Report 20/2021, exactly frozen UnsolvedMath record OWR-7155441-003, under either the printed index or logarithmic normalization. The n=2 case recovers the explicitly credited earlier paper DOI 10.5281/zenodo.23683405; the new result covers arbitrarily many factors at the same residue-field index. The fields and targets vary with n, the boundary points are not geometrically irreducible, and the base-field imperfection equals n. No claim is made about fixed-field unboundedness, base-field-imperfection thresholds, central-series assertions, corrected all-conjugate invariants or other characteristics. This four-page English unrefereed preprint was developed with AI assistance and self-audited. Exact polynomial checks supplement the written all-n proof; independent review, formal proof-assistant verification and absolute priority are not claimed.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-27
DOI
https://doi.org/10.5281/zenodo.23684181
Primary Topic
Algebraic Geometry and Number Theory
Type
preprint
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preprint

Normalized Maps of Arbitrary Arity to Wound Unipotent Groups

Alper Ferudun
2 links
Zenodo (CERN European Organization for Nuclear Research)
Algebraic Geometry and Number Theory
preprint

Normalized Maps of Arbitrary Arity to Wound Unipotent Groups

Alper Ferudun
preprint en

Abstract

For every integer n>=2, we construct a field k of characteristic two, a single closed point x of the projective line, and a nonconstant morphism from n copies of its complement to a smooth geometrically connected commutative k-wound unipotent group of dimension 2^n-1. The morphism is zero on every coordinate slice through infinity, while the residue field L always satisfies [L:kL^2]=2. The proof uses a determinant and the parameter-parity decomposition of a product of additive polynomials. For n>=3, this disproves the primitivity-bound assertion in Conjecture 5 of Oberwolfach Report 20/2021, exactly frozen UnsolvedMath record OWR-7155441-003, under either the printed index or logarithmic normalization. The n=2 case recovers the explicitly credited earlier paper DOI 10.5281/zenodo.23683405; the new result covers arbitrarily many factors at the same residue-field index. The fields and targets vary with n, the boundary points are not geometrically irreducible, and the base-field imperfection equals n. No claim is made about fixed-field unboundedness, base-field-imperfection thresholds, central-series assertions, corrected all-conjugate invariants or other characteristics. This four-page English unrefereed preprint was developed with AI assistance and self-audited. Exact polynomial checks supplement the written all-n proof; independent review, formal proof-assistant verification and absolute priority are not claimed.

Zenodo (CERN European Organization for Nuclear Research)
Algebraic Geometry and Number Theory
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