2025/10/19 by Louis Rowen, Rowen, Louis H., Uzi Vishne +1
Computer Science · Mathematics · #08B20 #16R20 #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Polynomial and algebraic computation #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2510.16825
openalex publication_date 2025/10/19 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
Suppose F is an infinite field and let f ∈ F\X1, …,Xm\ be a noncommutative polynomial. Partially answering a query of Makar-Limanov, we show that there are numbers d and m' such that, if F is closed under taking dth roots, for any n ≥ m' there are matrices A1,…,Am in~Mn(F) such that f(A1,…,Am) is upper triangular with n-m' prescribed diagonal entries. When f is homogeneous, f(A1,…,Am) is diagonal with n-m' prescribed diagonal entries. When f is multilinear, we can take d=1 and m' = [(m-1)/(2)], and the upper left (n-m')× (n-m') piece of f(A1,…,Am) can be taken to be diag(β1,…, βn-m'), for indeterminates βi. Furthermore, if f is not a polynomial identity of k × k matrices, then at least n - k characteristic values of f(A1,…,Am) may be taken to be algebraically independent.