Exercise 2, § 1

Let $A$ denote the ring $\mathbf{Z}[(X_n)_{n \in \mathbf{N}}, (Y_n)_{n \in \mathbf{N}}]$ of polynomials in the two families of indeterminates $(X_n)$ and $(Y_n)$, and $B$ the ring $\mathbf{Z}[X, Y]$ of polynomials in two indeterminates $X$ and $Y$. For every polynomial $R$ of $B$, there exists a unique sequence of polynomials $(R_n)_{n \in \mathbf{N}}$ of $A$ such that, for every $n \in \mathbf{N}$, one has

One has $R_0 = R(X_0, Y_0)$ and $R_n$ does not depend on the $X_i$ and $Y_i$ for $i > n$. If $R$ is homogeneous of degree $r$ with respect to $X$ (resp. $Y$, resp. $(X, Y)$) and if for every $i \in \mathbf{N}$ one assigns the weight $p^i$ to $X_i$ and $Y_i$, then $R_n$ is isobaric of weight $rp^n$ with respect to $(X_i)_{i \in \mathbf{N}}$ (resp. $(Y_i)_{i \in \mathbf{N}}$, resp. $((X_i)_{i \in \mathbf{N}}, (Y_i)_{i \in \mathbf{N}})$). If $R$ is constant, $R_n$ is constant. Examine the cases $R = 0, 1, -1, X + Y, XY, -X$.