Simple polynomial equations over (m x m)-matrices

• Chatyrko, Vitalij ^[1] ; Karassev, Alexandre ^[2]
  1. [1] Linköping University

     Linköping University

     Linköpings S:t Lars, Suecia

     
  2. [2] Nipissing University

     Nipissing University

     Canadá

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 27, Nº. 2, 2026
• Idioma: inglés
• DOI: 10.4995/agt.25707
• Enlaces
  • Texto completo
• Resumen

  • Let m be any integer ≥ 3 . We consider the polynomial equation             Xn+an-1·Xn-1+···+a1·X+a0·I=O,over (m x m)-matrices X with the real entries, where I is the identity matrix, O is the null matrix, ai  ∈ ℝ for each i and n ≥ 1. We discuss its solution set S supplied with the natural Euclidean topology.In particular, we describe the solution set S for m=3 and calculate its dimension.

• Referencias bibliográficas
  • V. A. Chatyrko and A. Karassev, Simple polynomial equations over (2 × 2)-matrices, Appl. Gen. Topol. 27, no. 1 (2025), 24202. https://doi.org/10.4995/agt.24202
  • R. Engelking, General Topology, Heldermann Verlag, Berlin, (1989)
  • D. Fuchs and A. Schwarz, A matrix Vieta Theorem, E. B. Dynkin Seminar, Amer. Math. Soc. Thansl. Ser. 2 169 (1996). https://doi.org/10.1090/trans2/169/02
  • S. Gelfand, On the number of solutions of a quadratic equation, In: Globus: General Mathematical Seminar, 1, Independent University of Moscow...
  • H. Lutkepohl, Handbook of Matrices, John Wiley & Sons (1996).
  • B. A. Pasynkov, On the coincidence of various definitions of dimensionality for factor spaces of locally bicompact groups, Uspekhi Matematicheskikh...
  • M. Slusky, Zeros of (2 × 2)-matrix polynomials, Comm. Algebra 38, no. 11 (2010), 4212-4223. https://doi.org/10.1080/00927870903366843
  • R. L. Wilson, Polynomial equations over matrices, Rutgers University, manuscript.