Consider the group ring $A$ of a free finitely generated group (i.e. noncommutative Laurent polynomials) with coefficients in a field $\mathbf{k}$ of characteristic $0$. Denote by $\tau: \: A \longmapsto \mathbf{k}$, the trace map given by the coefficient of the identity element of the free group (i.e. the monomial $\mathbf{1}$). If we consider $a \in A$ as a $(1 \times 1)$-matrix and for $n \geq 1$, $\tau(a^n)$ is the constant term of $a^n$. For any $a \in A$, the following formal series is algebraic \begin{equation} G =G(a):= \exp\Big(-\sum_{n \geq 1}\tau(a^n)\frac{t^n}{n}\Big)=1+... \in \mathbf{k}[[t]] \end{equation}
If we take, instead of the group algebra $A$, the set $M$ of square matrices of order $N$ with coefficients in $\mathbf{k}$, then we find $\tau$ as the usual trace map and $G$ as the characteristic polynomial in variable $t$ of a square matrix of $M$, \begin{equation} G=\det(1-ta),\: a \in M. \end{equation}
My problem is that I can’t compute $G$ in case $a=x_1+x_1^{-1}+x_2+x_2^{-1}+...+x_m+x_m^{-1}$.
Do you have any hints for the methodology or could you, if possible, indicate the first steps of the computation ? Can we use the residue formula ?
Many thanks.
PS; a conjecture by Maxim Kontsevich asks for a purely combinatorial proof (without references to number theory) of the algebraicity of $G$.