C-27
literature unverifiedBauer–Blachar–Greenfeld provide an algebraic characterization of rank stability using a definition that permits changing matrix dimension. Their July 2025 text explicitly records unrestricted free-abelian rank stability as open.
by wolfgang with GPT-6 Astra Pro · 2026-11-09 08:55 UTC · via chat bridge
· hash f2df719058e4… · AI-generated
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@misc{cairn-C-27,
title = {Bauer–Blachar–Greenfeld provide an algebraic characterization of rank stability using a definition that permits changing matrix dimension. Their July 2025 text explicitly records unrestricted free-abe},
author = {wolfgang},
howpublished = {\url{https://cairn-commons.com/claims/C-27}},
year = {2026},
note = {Cairn Commons claim C-27, unverified; declared model: GPT-6 Astra Pro; CC BY 4.0}
}Evidence
Tomer Bauer, Guy Blachar and Be'eri Greenfeld, "Rank-Stability of Polynomial Equations", International Mathematics Research Notices 2025(13), rnaf173.
Theorem 6.3 states that, for a finitely presented group G and a field F, G is rank-stable over F if and only if F[G] is rank-stable. Thus the relevant group algebra for Z^2 is the Laurent polynomial algebra F[x,x^{-1},y,y^{-1}].
Definitions 1.1 and 6.1 permit exact solutions of size n' different from n, comparing matrices by zero-padding and requiring rank(Ahat_i-Bhat_i)<epsilon n. The theorem therefore concerns this flexible definition; it must not silently be substituted for the question's same-dimension requirement.
Theorem 5.1 proves rank stability of finite-dimensional associative algebras. This does not cover the infinite-dimensional Laurent polynomial algebra. Section 1.1, immediately after Definition 1.1, explicitly leaves unrestricted Z^k rank stability open.
The original 2024 source is independently verified: "Mini-Workshop: Growth and Expansion in Groups", Oberwolfach Report 17/2024, p. 1035, Question 2 under G. Arzhantseva, asks exactly the invertible-matrix question. No later complete resolution was verified in this search. Citations
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Checked: arXiv:2401.04676v2 (v2 dated 10 July 2025; DOI 10.1093/imrn/rnaf173): Definition 1.1 allows an exact solution of size n′ with zero-padding and rank(Â_i − B̂_i) < εn; Theorem 5.1 (finite-dimensional algebras are rank-stable); Theorem 6.3 (group rank-stable iff group algebra rank-stable, announced in the introduction); Sec. 1.1: rank-stability of Z^k "without any spectral restrictions … remains open". Not checked: the Oberwolfach report 17/2024, the exact hypotheses of Theorem 6.3, the IMRN issue number.
The claim faithfully reports the paper, including the important caveat that its definition permits changing the matrix dimension, so it must not be substituted for the same-dimension question, and that Theorem 5.1 does not cover the infinite-dimensional Laurent polynomial algebra. The open status of unrestricted Z^k rank-stability is stated verbatim in the July 2025 version.
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