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<p>Motivated by recent developments in the Matrix Waring problems, we investigate the multiplicative version (proposed by M. Brešar and J. Volčič). Among our results, we prove that, under certain conditions, every square matrix over an algebraically closed division ring <i>D</i> can be expressed as a product of two images of a generalized polynomial over <i>D</i>, evaluated on square matrices over <i>D</i>. Our approach not only addresses this problem but also makes a modest contribution to the effort toward the generalized Lvov–Kaplansky conjecture (relating to (Linear Algebra Appl. 610: 827–836, 2021), while continuing Botha’s works (Linear Algebra Appl. 273: 65–82, 1998; Linear Algebra Appl. 286: 37–44, 1999) in more general setting of noncommutative division rings.</p>

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The Multiplicative Matrix Waring Problem for Algebraically Closed Division Rings

  • 期刊论文

  • 发表日期：

    2025年10月15日

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  • Truong Huu Dung,
  • Tran Nam Son

摘要

Motivated by recent developments in the Matrix Waring problems, we investigate the multiplicative version (proposed by M. Brešar and J. Volčič). Among our results, we prove that, under certain conditions, every square matrix over an algebraically closed division ring D can be expressed as a product of two images of a generalized polynomial over D, evaluated on square matrices over D. Our approach not only addresses this problem but also makes a modest contribution to the effort toward the generalized Lvov–Kaplansky conjecture (relating to (Linear Algebra Appl. 610: 827–836, 2021), while continuing Botha’s works (Linear Algebra Appl. 273: 65–82, 1998; Linear Algebra Appl. 286: 37–44, 1999) in more general setting of noncommutative division rings.