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arXiv · 2401.04676

Rank-stability of polynomial equations

Tomer Bauer·Guy Blachar·Be'eri Greenfeld
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Abstract

Extending the thoroughly studied theory of group stability, we study Ulam stability type problems for associative and Lie algebras; namely, we investigate obstacles to rank-approximation of almost solutions by exact solutions for systems of polynomial equations. This leads to a rich theory of stable associative and Lie algebras, with connections to linear soficity, amenability, growth, and group stability. We develop rank-stability and instability tests, examine the effect of algebraic constructions on rank-stability, and prove that while finite-dimensional associative algebras are rank-stable, `most' finite-dimensional Lie algebras are not.

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Tomer Bauer, Guy Blachar, Be'eri Greenfeld. 2025-07-10. Rank-stability of polynomial equations. https://doi.org/10.1093/imrn%2Frnaf173

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