I've been doing some reading on cluster algebra from surfaces and in Schiffler I encountered this Definition:
Let $\mathcal{X}$ be the set of all cluster variables obtained by mutation from $(\textbf{x}, \textbf{y}, Q)$. The cluster algebra $\mathcal{A} = \mathcal{A}(\textbf{x}, \textbf{y}, Q)$ is the $\mathbb{ZP}$-subalgebra of $\mathcal{F}$ generated by $\mathcal{X}$.
By definition, the elements of $\mathcal{A}$ are polynomials in $\mathcal{X}$ with coefficients in $\mathbb{ZP}$, so $\mathcal{A} ⊂ \mathbb{ZP}[\mathcal{X} ]$. On the other hand, $\mathcal{A} \subset \mathcal{F}$, so the elements of $\mathcal{A}$ are also rational functions in $x_{1},\dots , x_{n}$ with coefficients in $\mathbb{QP}$.,
where $\mathbb{P}$ is a tropical semifield, $\mathbb{ZP}$ is its group ring and $\mathcal{F}=\mathbb{QP}(x_{1},\dots , x_{n})$ is the field of rational functions in $n$ variables and coefficients in $\mathbb{QP}$.
I'm not sure about the part when we say that the elements of $\mathcal{A}$ are polynomials. For instance, looking at the elements of the cluster algebra from this example , how would one see them as polynomials? Is this just about introducing some new abbreviations for some of the cluster variables (if so, how do we pick the 'right' cluster variables?)? So that we can write every (other) cluster variable as a polynomial expression in those new abbreviations (can we always do that)? Any hints or explanations are much appreciated.