Noncommutative algebraic equations and the noncommutative eigenvalue problem (Q1590372)

For the noncommutative algebraic equations \[ x^n= a,\quad x^{n-1}+ a_2x^{n-2}+\cdots+ a_n\tag{1} \] the noncommutative eigenvalue \((\lambda)\) problem: \(AX= \lambda X\) \((A= (a_{ij}), \widetilde X= (x_j))\), i.e. the generalized Vieta theorem [cf. \textit{D. Fuchs} and \textit{A. Schwarz}, Am. Math. Soc. Transl., Ser. 2, 169, 15-22 (1995; Zbl 0837.15011); \textit{A. Connes} and \textit{A. Schwarz}, Lett. Math. Phys. 39, No. 4, 349-353 (1997; Zbl 0874.15010)] is analyzed. As a result, the theorem is proved about the structure of perturbation series for the traces Tr\(x^r\), where \(x\) is solution of the equation (1). This theorem generalizes to the case of arbitrary number \(r\) the recently proposed \((r=1)\) perturbative approach applied to the gauge \(U(1)^k\)-invariant Born-Infeld Lagrangian [cf. \textit{P. Aschieri}, \textit{D. Brace}, \textit{B. Morariu} and \textit{B. Zumino}, Nucl. Phys. B 588, No. 1-2, 521-527 (2000; Zbl 0972.81092)]. The noncommutative generalization of the Vieta theorem is considered as a part of the general theory of noncommutative functions [cf. \textit{I. Gelfand} and \textit{V. Retakh}, The Gelfand Mathematical Seminars, 93-100 (1996; Zbl 0865.05074); Sel. Math., New Ser. 3, No. 4, 517-546 (1997; Zbl 0919.16011)].