References:
Introduction to Tropical Geometry, Diane Maclagan, Bernd Sturmfels
Tropical Analytic Geometry, Newton Polygons, and Tropical Intersections, Joseph Rabinoff
Uniform Bounds for the Number of Rational Points on Symmetric Squares of Curves with Low Mordell-Weil Rank, Sameera Vemulapalli and Danielle Wang
Let \(P\subset N_\mathbb{R}\) be a polyhedron. The cone of unbounded directions or recession cone of \(P\) is defined as the cone \(\mathcal{U}(P)\)
\[\mathcal{U}(P) = \{v\in N_\mathbb{R}\mid P+tv\subset P \text{ for all } t\geq 0\}.\]
Alternatively, it is also the dual of the cone
\[\mathcal{U}(P)^\vee = \{u\in M_\mathbb{R}\mid \mathrm{face}_u(P)\neq\emptyset\}.\]
Here you either use the min set for the face and \(\ge\) for dual, or max set for the face and \(\le\) for dual. The two definitions are equivalent.
Let \(\sigma\subset N_\mathbb{R}\) be an integral pointed cone, and \(K[S_\sigma]\) will be the ring. We have
\[K[S_\sigma] = \left\{\sum_{u\in S_\sigma} a_u x^u : a_u, \text{ almost all zero}\in K\right\}.\]
and \(X(\sigma) := \mathrm{Spec}(K[S_\sigma])\).
The polyhedral subdomain associated to \(P\) is the set \(U_P=\mathrm{trop}^{-1}(\overline{P})\subset X(\sigma)^{an}\)
Let \(P\subset N_\mathbb{R}\) be an integral \(\Gamma\)-affine pointed polyhedron with cone of unbounded directions \(\sigma = \mathcal{U}(P)\), define
\[K\langle U_P\rangle= \left\{\sum_{u\in S_\sigma} a_u x^u : \mathrm{val}(a_u)+\langle u,p\rangle \to 0,\,\forall p\in P \right\} .\]
Here we have to restrict to terms that lies in \(S_\sigma\) because with these terms it is impossible to converge, \(S_\sigma\) is the most general form you can get. Now you still need the general term to go to zero.
Idea. Think of \(K\langle U_P\rangle\) as the set of functions that converges on valuations in the polyhedron \(P\).
The ring \(K\langle U_P\rangle\) is a \(K\)-affinoid algebra.
The obvious inclusion \(K[S_\sigma]\hookrightarrow K\langle U_P\rangle\) induces an open immersion
Let \(f_1,\dots,f_d\in K\langle \mathcal{U}(P)\rangle, Y_i = V(f_i)\) and \(Y=\cap_{i=1}^d Y_i\). Assume \(Y\) is zero-dimensional. Then the intersection multiplicity of \(Y_1,\dots,Y_d\) at \(w\in Y\) is defined as
\[i(w,Y_1,\cdots,Y_d) = \dim_K H^0(Y\cap \mathcal{U}_{\{w\}}, \mathcal{O}_{Y\cap \mathcal{U}_{\{w\}}}).\]
when it is finite, it is
\[i(w,Y_1,\cdots,Y_d) = \sum_{\mathrm{trop}(\xi)=w} \dim_K \mathcal{O}_{Y,\xi}\]
which counts the number of common zeros of \(f_i\) having the same valuation as \(\{w\}\) with multiplicity.
For \(f\in K\langle \mathcal{U}(P)\rangle\), the height graph of \(f\) is the set
\[H(f) = \{(u, \mathrm{val}(a_u)) : u\in S_P, a_u\neq 0\}.\]
There is an obvious projection map that forgets the valuation \(\pi: \mathbb{R}^n\times\mathbb{R}\to \mathbb{R}^n\).
This is actually a high dimensional generalization of \(p\)-adic newton polygon.
Associated with the height graph, we define
\[m_w(f) = \min [ (w,1)\cdot H(f) ].\]
\[\mathrm{vert}_w(f) = \{h\in H(f) : (w,1)\cdot h = m_w(f)\}.\]
Example.
Consider the ordinary theory of \(p\)-adic Newton polygon here, if \(f\in \mathbb{Q}_p[t]\), then \(H(f)\) gives the points whose lower convex hull is the Newton polygon of \(f\).
Given input valuation \(v(t)=w\), the product \((w,1)\cdot (u,v(a_u)) = wu + v(a_u) = v(a_u t^u)\). So \(\mathrm{vert}_w(f)\) is finding the terms in \(f\) that achieves the minimum valuation (and thus the maximum norm) at input valuation \(w\).
Let \(P_1,\dots,P_d\) be bounded polyhedra in \(\mathbb{R}^d\). Define the function
\[V_{P_1,\dots,P_n}(\lambda_1,\dots,\lambda_d) = \mathrm{vol}(\lambda_1 P_1 + \cdots + \lambda_d P_d)\]
where \(\lambda_i P_i\) is multiplying vector by scalar, and \(+\) is the Minkowski sum. The mixed volume of \(P_1,\dots,P_d\) is the coefficient of the \(\lambda_1\dots \lambda_d\) term of \(f\),
\[\mathrm{MV}(P_1,\dots,P_d) = V_{P_1,\dots,P_d}(\lambda_1,\dots,\lambda_d)[\lambda_1\cdots \lambda_d].\]
Example. Consider \(P_1 = [0,a]^2\) and \(P_2 = [0,b]^2\). Then
\[\begin{align*} V_{P_1,P_2}(\lambda_1,\lambda_2) &= \mathrm{vol}(\lambda_1 P_1 + \lambda_2 P_2)\\ &= \mathrm{vol}([0,\lambda_1a+\lambda_2b]^2) \\ &= (\lambda_1a+\lambda_2b)^2 \\ &= \lambda_1^2a^2 + 2ab\lambda_1\lambda_2 + \lambda_2^2b^2 \end{align*}\]
We see from the coefficient of \(\lambda_1\lambda_2\) that the mixed volume is \(\mathrm{MV}(P_1,P_2) = 2ab\).
Remark. The definition is translation invariant, it does not matter where you put these polyhedra. \(V\) will be a homogeneous polynomial, the mixed volume is also multi-linear when passed to constant scaling on polyhedron.
Theorem. Let \(f_1,\dots,f_d\in K[x_i^{\pm}]_{i}\) be Laurent polynomials with finitely many common zeros. Then the number of common zeros counted with multiplicity in \((K^\times)^d\) is bounded by
\[\mathrm{MV}(\mathrm{NP}(f_1),\dots,\mathrm{NP}(f_d)).\]
This theorem does not make use of much of the information though, because it does not utilize the valuation (Newton polygon here does not use the height graph). But nevertheless it seems interesting as a kind of degree bound of Laurent polynomials.
Theorem (Katz, Osserman-Payne). Let \(f_1,\dots,f_d\in K[M]\) and let \(v\in \cap_{i=1}^n \mathrm{trop}(f_i)\) be an isolated point. Define \(Y_i=V(f_i)\) and \(\gamma_i(v)=\pi(\mathrm{conv}(\mathrm{vert}_v(f_i)))\) be the corresponding polytope (think as a ‘slope’). Then
\[i(v,Y_1,\dots,Y_d) = \mathrm{MV}(\gamma_1,\dots,\gamma_d).\]