Surjectivity of polynomial maps on Matrices

2023/05/31 by Saikat Panja, Panja, Saikat, Prachi Saini +3 · 2 citations
Mathematics · Physics and Astronomy · #11P05 #16S50 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2305.19731

openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For n≥ 2, we consider the map on M_n(\mathbb K) given by evaluation of a polynomial f(X₁, …, X_m) over the field \mathbb K. In this article, we explore the image of the diagonal map given by f=δ₁ X₁^k₁ + δ₂ X₂^k₂ + ⋯ +δ_m X_m^k_m in terms of the solution of certain equations over \mathbb K. In particular, we show that for m≥ 2, the diagonal map is surjective when (a) \mathbb K= \mathbb C, (b) \mathbb K= \mathbb F_q for large enough q. Moreover, when \mathbb K= \mathbb R and m=2 it is surjective except when n is odd, k₁, k₂ are both even, and δ₁δ₂>0 (in that case the image misses negative scalars), and the map is surjective for m≥ 3. We further show that on M_n(\mathbb H) the diagonal map is surjective for m≥ 2, where \mathbb H is the algebra of Hamiltonian quaternions.

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• Dynamics of Word Maps on Groups and Polynomial Maps on Algebras
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