A proof of Kapranov’s Theorem in Tropical Geometry

Kapranov’s Theorem is a simpler version of the Fundamental Theorem of Tropical Algebraic Geometry. It states that for any nice Laurent polynomial $f$, the tropical hypersurface $trop(V(f))$, the set $\{w \in \mathbb{R}^n : \text{initial form of } w \text{ is not a monomial}\}$, and the closure of $\{(val(y_1),…,val(y_n)) : (y_1,…,y_n) \in V(f)\}$ all coincide. I will be working through the background knowledge needed to understand Kapranov’s Theorem and its proof, and present the proof in detail.