10 1. THE TROPICS 0 1 0 0 Figure 12. The Newton polytope and subdivision for Figure 5. give the slopes of ˇ = f on the two domains of linearity of f on either side of ˇi, so ni must be constant on ˇi. Obviously, we have (1.1) k i=1 wini = 0. We call this the balancing condition. In the case when dim MR = 2, so that V (f) is a curve, it is useful to rewrite this as follows. Let V ˇ[0] be a vertex of V (f), contained in edges E1,...,Ek ˇ[1] of V (f), and let m1,...,mk M be primitive tangent vectors to E1,...,Ek pointing away from V . Suppose Ei has weight wi. Then (1.1) is equivalent to (1.2) k i=1 wimi = 0. Example 1.6. The tropical ezout theorem. Suppose dim MR = 2, and let e1,e2 be a basis for M. Let Δd be the polytope which is the convex hull of 0,de1, and de2. If f = n∈Δd anzn, then V (f) is a tropical curve in MR, which we call a degree d curve in the tropical projective plane. For example, Figure 2 depicts a degree 1 curve, i.e., a tropical line, and Figures 3 and 4 depict degree 2 curves, i.e., tropical conics. These should be thought of as tropical analogues of ordinary lines and conics in P2. These tropical versions often share surprising properties in common with the usual algebraic versions. We give one example here. Let C, D MR be two tropical curves in the tropical projective plane of degree d and e respectively. Suppose that C and D intersect at only a finite number of points this can always be achieved by translating C or D. In fact, we can similarly assume that none of these intersection points are vertices of C or D. We can define a notion of multiplicity of an intersection point of these two curves. Suppose that a point P C D is contained in an edge E of C and an edge F of D, of weights w(E) and w(F ) respectively. Let m1 be a primitive tangent vector to E and m2 be a primitive tangent vector to F . Then we define the intersection multiplicity of C and D at P to be the positive integer iP (C, D) := w(E)w(F )|m1 m2|.
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