In his paper "Noncommutative identities" M. Kontsevich states the following:
Theorem 2. For any $P=P(x,y)=1+\cdots\in\mathbb{C}[x,y]$ expand $$\log(P)=\sum_{n,m}f_{n,m}x^ny^m\in\mathbb{C}[[x,y]].$$ Then the series $$\exp\left(\sum_{n,m}{n+m \choose n}f_{n,m}x^ny^m\right)$$ is algebraic.
The hint he gives is to use the residue formula twice. I've wasted hours and hours trying to make sense of this statement, but none of my efforts have led anywhere, the main one being: \begin{align*} \sum_{n,m}{n+m \choose n}f_{n,m}x^ny^m & =\sum_{n,m}\text{"Tr"}\bigg((X+1)^n(X^{-1}+1)^m\bigg)f_{n,m}x^ny^m \\ & = \text{"Tr"}\left(\sum_{n,m} f_{n,m}\bigg((X+1)\cdot x\bigg)^n\bigg((X^{-1}+1)\cdot y\bigg)^m\right) \\ & =-\text{"Tr"}\left( \sum_{k\ge1}\frac{Q^k\bigg((X+1)\cdot x,(X^{-1}+1)\cdot y\bigg)}{k} \right) \\ & =- \sum_{k\ge1}\frac{\text{"Tr"}\left( Q^k\bigg((X+1)\cdot x,(X^{-1}+1)\cdot y\bigg) \right)\tau^k}{k}\bigg|_{\tau=1} \end{align*}
Where $"\text{Tr}"$ stands for the constant coefficient and X is a (noncommutative) formal variable. Then applying the following construction: \begin{align*} \exp\left(-\sum_{k\ge1}\frac{\text{"Tr"}(a^k)}kt^k\right) & = \exp\left(-\sum_{k\ge1}\frac{\text{res}[a^k/z,0]}k t^k\right) = \exp\left(-\frac1{2\pi i}\sum_{k\ge1}\oint_{|z|=\epsilon}\frac{a^k(z)}k t^k\frac{dz}{z}\right)\\ & = \exp\left(\frac1{2\pi i}\oint_{|z|=\epsilon}\left(-\sum_{k\ge1}\frac{a^k(z)t^k}k d\right)\frac{dz}{z}\right) \\ & = \exp\left(\frac1{2\pi i}\oint_{|z|=\epsilon}\frac{\log(1-a(z)t)}zdz\right) \\ & = \exp\left(\int \left( \frac1{2\pi i} \oint_{|z|=\epsilon}\frac{-a(z)}{1-a(z)t}\frac{dz}{z}\right)dt\right)\\ \end{align*}
But then looking for the poles of this function in the neighbourhood of the origin doesn't quite work... (for "classical" Laurent polynomials in one variable (i.e with coefficients only in $\mathbb{C}$ instead of $\mathbb{C}[x,y]$) this worked quite well)
Therefore here I am, looking for any piece of advice to get closer to a palpable result. Many thanks in advance!
PS: The whole paper can be found here: http://arxiv.org/pdf/1109.2469v1.pdf