Polynomial Maps with Constants over Division Algebras and the Generalized Kaplansky--L'vov Conjecture

2026/07/10 by Archit Gangwal, Arunava Mandal, Somesh Verma
Mathematics · #math.RA #math.GR #msc:16K20 #msc:16S50

paper · pdf

15 pages, o figure

arxiv created 2026/07/10 · arxiv updated 2026/07/31

Abstract

The Kaplansky--L'vov conjecture asserts that the image of a multilinear polynomial map on a full matrix algebra over a field is always a vector space. Although the conjecture remains open in general, substantial progress has been made for 2× 2 and 3× 3 matrix algebras over various fields. Recently, Panja, Saini, and Singh formulated a generalized Kaplansky--L'vov conjecture for polynomial maps with matrix coefficients over algebraically closed fields and verified it for 2× 2 matrices. In this work, we investigate an analogous problem for polynomial maps with constant matrix coefficients over an infinite division algebra. Specifically, we consider polynomials in the free algebra M₂(\mathbb D)⟨ x₁,…,x_m⟩ of the form ω= A₁x₁^k₁+⋯+A_mx_m^k_m, where the A₁,…, A_m∈ M₂(\mathbb D) are fixed matrices, \mathbb D is an infinite division algebra, and k₁,…, k_m are positive integers. We prove that the corresponding generalized Kaplansky--L'vov conjecture holds for 2× 2 matrices over \mathbb R and the quaternion division algebra \mathbb H. We also investigate the surjectivity of these polynomial maps. This can be viewed as a generalized Waring problem for matrix algebras.

Citations

• Images of polynomial maps with constants
• Word maps, word maps with constants and representation varieties of\n one-relator groups

Related

• The L'vov-Kaplansky Conjecture for Polynomials of Degree Three
• The images of non-commutative polynomials evaluated on 2× 2 matrices over an arbitrary field
• Image of arbitrary polynomials on upper triangular matrix algebras
• The images of non-commutative polynomials evaluated on the Quaternion algebra
• Images of polynomial maps with constants