I am well aware of this. If you look at my comments within this very chain, you will see that I made reference to polynomial evaluation, which I will call Ev(f, x). I Am not denying that the partial application given by f' = x -> f(f,x) is naturally induced by the polynomial f. Nor that this is so natural that it often makes sense to identify f with f' so that we would consider f=f', even though they are different types of objects.

>Ask a million mathematicians, “is x^2+3x a polynomial” and without hesitation they’ll say yes.

As will I, because the distinction between formal polynomials and expressions which can be naturally modeled as polynomials is so unimportant that it is almost never worth thinking about.

Put another way, how would you compute the following sets:

{ x \in C | x^3 - x = 8 }

{ x \in C | x^3 = x + 8 }

{ x \in C | log(x) = x^x }

{ x \in C | sin(x) = .7 }

{ x \in N | exists y \in N such that 5x + 3y = 1 }

Are you really claiming that these questions are ill-posed without stating them in terms of algebraic geometry?