Properties of perturbative solutions of unilateral matrix equations (Q5943723)

A left-unilateral matrix equation is an algebraic equation of the form \(a_0+a_1x+a_2x^2+\ldots +a_nx^n=0\) where the coefficients \(a_r\) and the unknown \(x\) are square matrices of the same order and all coefficients are on the left (similarly for a right-unilateral equation). The paper studies the unilateral matrix equations \[ x^n=1+\varepsilon (a_0+a_1x+\ldots +a_{n-1}x^{n-1}) \] and \[ \Phi =A_0+A_1\Phi +\ldots +A_n\Phi ^n \] and the relation between their solutions. Explicit formulas (which are matrix series) are obtained for Tr\(x^s\) and Tr\(\Phi ^s\), \(s\in {\mathbf N}\). The coefficients in these formulas depend on symmetrized polynomials in the coefficients \(a_r\) (resp. \(A_r\)). To obtain the formulas the authors express the trace of the solution as a contour integral in the complex plane of the trace of the resolvent of the corresponding matrix, then use the basic property of the trace of the logarithm of matrices and the generalized Bézout theorem. The latter states that if \[ P(x)=a_0+a_1x+a_2x^2+\ldots +a_nx^n, \] then \[ P(\lambda)-P(x)=Q(\lambda ,x)(\lambda -x) \] where \[ Q(\lambda ,x)=\sum _{l=0}^{n-1}\lambda ^l(\sum _{r=l+1}^na_rx^{r-l-1}), \] i.e. \(\lambda -x\) is a divisor of \(P(\lambda)-P(x)\) on the right (if the coefficients are to the right, then it would be a divisor to the left). The paper continues the ideas of \textit{P. Aschieri, D. Brace, B. Morariu}, and \textit{B. Zumino} [Nucl. Phys. B 588, No. 1-2, 521-527 (2000; Zbl 0972.81092)] and of \textit{A. Schwarz} [Lett. Math. Phys. 52, No. 2, 177-184 (2000; Zbl 0972.15001)].