I'm looking for the name of a "division-like" operation $f$ such that $f(P,x^n)$ walks over the $x$-rooted terms of the polynomial $P$ and performs integer division on each exponent, dividing by $n$, discarding remainders, so e.g. $f(x^{256},x^2)=x^{128}$ and $f(x^{11},x^2)=x^5$. And for example given $P=x+2x^2+3x^3+2x^4+x^5$, then $f(P,x^2)=1+5x+3x^2$, and $f(P,x^3)=3+6x$.
There should be an identity $f(Q,x)=Q$ for any polynomial $Q$ involving the variable $x$.
The operation should also distribute over addition, so $f(P,x+x^n+x^m)$ should equal $P+f(P,x^n)+f(P,x^m)$.
Does this operation have a name?
It is a linear map on the vector space of polynomials.
i.e.
$f(P,x^2) = \begin{bmatrix} 1&1\\&&1&1&\\&&&&1&1\\\end{bmatrix} \begin{bmatrix}0\\1\\2\\3\\2\\1\end{bmatrix} = \begin{bmatrix}1\\5\\3\end{bmatrix}$
No, this doesn't have a name as far as I am aware:
$f(P,x^3) = \begin{bmatrix}1&1&1&\\&&&1&1&1\\\end{bmatrix} \begin{bmatrix}0\\1\\2\\3\\2\\1\end{bmatrix} = \begin{bmatrix}3\\6\end{bmatrix}$