Given the tropical semiring $(\mathbb{T},\oplus,\otimes)$, a tropical hypersurface associated to a tropical polynomial is the set of points where it is non-differentiable.
I'm wondering how incidental the non-differentiable part is, and how this part is derived. For instance, had we given a different semiring $S$, would there be a canonical way of associating to $S$-polynomials $S$-hypersurfaces? I imagine this might have something to do with Universal Algebra, of which I am ignorant, so it would be good to get a light answer on this as much as possible, and if not possible a light reference would be greatly appreciated!
On not initially receiving an answer, I did some further reading on this and found a lovely light exposition: https://www.mathenjeans.fr/sites/default/files/documents/Sujets2012/tropical_geometry_-_casagrande.pdf
The idea is to construct the tropical semiring $(\mathbb{T},\oplus,\otimes)$ in analogy to the reals via closure of operations from the more primitive $(\mathbb{N},\oplus,\otimes)$. What a root is turns out to be a pivotal concept. So the root of an $\mathbb{R}$-monomial $P(x) = ax+b$ is a number $x_0$ for which $P(x_0)=0$, the additive identity. If we try transferring this over naively, we get the following: $$ \begin{align} a \otimes x \oplus b &= -\infty \\ \iff \max(a+x,b) &= - \infty, \end{align} $$ which is not analogously informative. In the above paper, they provide an analogue of the fundamental theorem of algebra, and give a justification for the 'non-differentiability' aspect as a consequence of wanting our $\mathbb{T}$-roots to give us the information on the factorisation implied by the fundamental theorem.
So the associated hypersurface of a tropical polynomial is its set of tropical roots as defined above. Very happy to accept answers that might go into more detail or on anything I missed here!