Polynomial maps on indeterminate of vector space of polynomials I am studying polynomial rings and would like to get an idea of what it takes to study the problems of transforming polynomial forms by performing polynomial map on indeterminate of the polynomials.
Given a polynomial ring $V$, if I perform a transformation on indeterminate, and consider only polynomial function of transformation, e.g., denote the indeterminate as $x$, do $x = p(y)$, where $p$ is a polynomial function, then we performed a linear map from $V$ to another polynomial ring, say $W$, with indeterminate $y$ (although the specific choice of symbol is not important). Specifically, I am satisfied with results only over polynomial rings defined on $\mathbb{R}$, with $deg(p)\le d$, i.e., finite dimensional polynomial rings. However, I am also interested in specific subspaces, for example, if I understand correctly, choose any specific monomials of indeterminate to form a basis, you can generate a vector space.
My goal is to investigate such questions, given any two specific vector spaces of polynomials, say $V$ and $W$, can I find such a polynomial map on indeterminate to map V to a subspace of $W$? First question might be, can any linear map between two vector space of polynomials be represented by such a polynomial map on indeterminate? Ultimately I want to be able to investigate questions such as, given $n$ indeterminate ${x_{1}, x_{2}, ..., x_{n}}$, form a basis with certain monomials of them with degree no larger than d, and generate a vector space $V$, repeat similar procedure to get another vector space $W$ (with possibly a different dimension of indeterminate), does there always exist a linear map that can be represented as polynomial maps on indeterminate that transform V to a subspace of $W$, and how can I find it? (Of course the dimensionality of $W$ is no less than that of $V$)
I hope to be introduced some basic techniques and ways of thinking it. Many thanks!
 A: You seem to be specifically interested in maps $K[X]\to K[Y]$ defined by the substitution of some polynomial $P\in K[Y]$ for $X$. This indeed defines a linear map $K[X]\to K[Y]$, but it is much more than that, it is also a morphism of rings. That means that there are many linear maps (those that are not morphisms of rings) that you cannot obtain in this way. On the other hand you can get all $K$-linear morphisms of rings $K[X]\to K[Y]$ this way, as such a map is determined by its image of $X$, and it suffices to take for $P$ that image. Studying maps defined by polynomials is an important but vast subject, ultimately leading to the subject area called commutative algebra. The point of view of considering these maps just as $K$-linear maps looses a lot of information, so it is not often very useful. However see this answer for an example where it is helpful.
A: Any linear map between vector spaces is a polynomial map: actually a homogeneous polynomial of degree 1, hence the name linear map. This answers your question in the third paragraph. (your word indeterminate possibly means an element of the  basis of the dual vector space).
You can possibly read about symmetric algebra of a vector space to get clarity on these notions.
