We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.2000 Mathematics Subject Classification. 13P10, 14Q99, 52B20, 68W30.

Abstract. A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve Γ analogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from Γ to a tropical projective space, and the image can be extended to a tropical curve of degree equal to deg(D). The tropical convex hull of the image realizes the linear system |D| as a polyhedral complex. We show that curves for which the canonical divisor is not very ample are hyperelliptic. We also show that the Picard group of a Q-tropical curve is a direct limit of critical groups of finite graphs converging to the curve.

Abstract. Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals which generalize the hull complex of Bayer and Sturmfels [1], instances of which improve upon the hull resolution in the sense of being smaller. We also suggest a new definition of a face of a tropical polytope, which has nicer properties than previous definitions; we give examples and provide many conjectures and directions for further research in this area.

The tropical convex hull of a finite set of points in tropical projective space has a natural structure of a cellular free resolution. Therefore, methods from computational commutative algebra can be used to compute tropical convex hulls. Tropical cyclic polytopes are also presented.

Abstract. The hyperdeterminant of format 2 × 2 × 2 × 2 is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4-cube. The 87959448 regular triangulations of the 4-cube are classified into 25448 Dequivalence classes, one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 of them come from the hyperdeterminant.