Trying to understand the proof of Laurent phenomenon of cluster algebras

Question

I am trying to understand the proof of Laurent phenomenon of cluster algebras in the book (Sergey Fomin, Lauren Williams, Andrei Zelevinsky, Introduction to Cluster Algebras. Chapters 1-3, arXiv:1608.05735v1).

On page 45, it is said that "We see that $x_j'$ is linear in $x_q$ and hence irreducible (as a Laurent polynomial in $\tilde{\bf x}$), i.e., it cannot be written as a product of two non-monomial factors." Here \begin{align} x_j' = x_j^{-1} (x_k^c x_q M_1 + M_2), \end{align} where $M_1, M_2$ are monomials in $x_i$'s ($i \not\in \{j,k,q,r\}$).

Why $x_j'$ cannot be written as a product of two non-monomial factors. Maybe we can have something like \begin{align} x_j' = (x_q+x_r)(x_j+1)? \end{align} Here $(x_q+x_r)(x_j+1)$ is linear in $x_q$ and $(x_q+x_r)(x_j+1)$ is a product of two non-monomial factors.

Where do I made a mistake? Thank you very much.

Answer 1

You need to use the fact that $x'_j = M x_q + M'$, where $M$ and $M'$ are Laurent monomials in the remaining variables. Their observation is that if $x'_j$ factored, it would be as $x'_j = (P_1 x_q + P_2)P_3$, where $P_1, P_2, P_3$ are Laurent polynomials in the remaining variables. But then $P_1 P_3 = M$ and $P_2 P_3 = M'$, which forces each of $P_1, P_2, P_3$ to be Laurent monomials in the remaining variables. So there is no non-trivial factorization of $x'_j$.