Notation for polynomials of polynomials

Question

Let $q(x) = x^2$ and $p(x, y) = x^2 + y^2 = q(x) + q(y)$. Then, I can write $p \in \mathbb{R}[x, y]$, but I would like to write something like $p \in \mathbb{R}[q(x), q(y)]$. Is such a thing valid notation? And if it is, does it change the definition of $p$, i.e., would then $p(x, y) = x + y$?

Answer 1

Let $q(x) = x^2$. Then you can write $\mathbb{R}[q(x), q(y)]=\mathbb{R}[x^2, y^2]$, and this would mean the ring of polynomials over $\mathbb{R}$ in which only even powers of $x$ and $y$ appear.

Then $p(x, y) = x^2 + y^2 \in\mathbb{R}[x^2, y^2]$ since the only powers of $x$ and $y$ which appear in $p$ are even.

Answer 2

The notation $\mathbb{R}[q(x), q(y)]$ is always posible if you fixe the polynomial $q(x)$ but what you get is a (strict) subset of $\mathbb{R}[x,y]$ (in fact well structured, is it a subring?). With this you discard infinitely many polynomials of $\mathbb{R}[x,y]$. Anyway your notation could be useful for certain particular goals.