[gizmo686]

> I believe you are the only person trained in mathematics in the world that thinks x^2+3x=0 is not a polynomial equation.

Then you are not reading my comments.

> To be clear, outside of very particular contexts I would still call x^2-x+1=0 a polynomial equation, because it is extremely useful to talk about polynomials without invoking all of the machinery of formal polynomials.

>You don’t understand the underlying algebraic theory.

My claim is that the underlying algebraic theory is not necessary to rigorously state or solve the problems under discussion.

>You can’t write such a statement if you really understand that “what values make x+3 the number 5” is a nicer way of conveying the question “under the natural evil map induced by x+3 what is the pre-image of 5”.

My claim is that "what values make x+3 the number 5" translates directly into { x \in F | x+3=5}. I further claim that the "x+3" in this interpenetration is not a polynomial in the formal sense

I am not disputing that this is set is the same set as the pre-image of 5 under the natural map induced by x+3.