Every tropical variety is the intersection of a finite number of tropical hypersurfaces. A finite set of polynomials is called a tropical basis for X if is the intersection of the tropical hypersurfaces of . In general, a generating set of is not sufficient to form a tropical basis. The intersection of a finite number of a tropical hypersurfaces is called a tropical prevariety and in general is not a tropical variety. Initial ideals Choosing a vector in defines a map from the monomial terms of to by sending the term m to . For a Laurent polynomial , define the initial form of f to be the sum of the terms of f for which is minimal. For the ideal , define its initial ideal with respect to to be Then define Since we are working in the Laurent ring, this is the same as the set of weight vectors for which does not contain a monomial. When K has trivial valuation, is precisely the initial ideal of with respect to the monomial order given by a weight vector . It follows that is a subfan of the Gröbner fan of . Image of the valuation map Suppose that X is a variety over a field K with valuation v whose image is dense in (for example a field of Puiseux series). By acting coordinate-wise, v defines a map from the algebraic torus to . Then define where the overline indicates the closure in the Euclidean topology. If the valuation of K is not dense in , then the above definition can be adapted by extending scalars to larger field which does have a dense valuation. This definition shows that is the non-Archimedean amoeba over an algebraically closed non-Archimedean field K. If X is a variety over , can be considered as the limiting object of the amoeba as the base t of the logarithm map goes to infinity. Polyhedral complex The following characterization describes tropical varieties intrinsically without reference to algebraic varieties and tropicalization. A set V in is an irreducible tropical variety if it is the support of a weighted polyhedral complex of pure dimension d that satisfies the zero-tension condition and is connected in codimension one. When d is one, the zero-tension condition means that around each vertex, the weighted-sum of the out-going directions of edges equals zero. For higher dimension, sums are taken instead around