I've seen people refer to $n$th order linear differential equations as an "$n$th order linear polynomial of the function $y$ and its derivatives."
I can see how it definitely share the same form as a polynomial:
I also see this in how people characterize homogeneous LDE's as such because they are homogeneous polynomials (i.e all the terms equal $0$). I get the analog being drawn but these aren't truly polynomials right? How can we say these are polynomials if they are not being exponentiated?
Is there a more general notion of polynomial or '$n$th order' that would encompass $n$th order differential equations and $n$th order polynomials?