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We give several characterizations of stable intersections of tropical cyclesand establish their fundamental properties. We prove that the stableintersection of two tropical varieties is the tropicalization of theintersection of the classical varieties after a generic rescaling. A proof ofBernstein's theorem follows from this. We prove that the tropical intersectionring of tropical cycle fans is isomorphic to McMullen's polytope algebra. Itfollows that every tropical cycle fan is a linear combination of pure powers oftropical hypersurfaces, which are always realizable. We prove that every stableintersection of constant coefficient tropical varieties defined by prime idealsis connected through codimension one. We also give an example of a realizabletropical variety that is connected through codimension one but whose stableintersection with a hyperplane is not.

Stable Intersections of Tropical Varieties