We study a tropical analogue of the projective dual variety of a hypersurface. When @math is a curve in @math or a surface in @math , we provide an explicit description of @math in terms of @math , as long as @math is smooth and satisfies a mild genericity condition. As a consequence, when @math is a curve we describe the transformation of Newton polygons under projective duality, and recover classical formulas for the degree of a dual plane curve. For higher dimensional hypersurfaces @math , we give a partial description of @math .
In a slightly different setting, B. Bertrand, E. Brugall 'e, and B. Mikhalkin study of tropical cycles in the plane to tropical morphisms @cite_13 . of tropical plane curves have been studied by the second author together with M. Baker, R. Morrison, N. Pflueger, and Q. Ren @cite_11 , H. Markwig @cite_2 , and H. Lee @cite_12 .
@cite_2 @cite_13 @cite_12 @cite_11
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We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic. This paper is about the combinatorics of finite point configurations in the tropical projective space or, dually, of arrangements of finitely many tropical hyperplanes. Moreover, arrangements of finitely many tropical halfspaces can be considered via coarsenings of the resulting polyhedral decompositions of Rd. This leads to natural cell decompositions of the tropical projective space TPmind−1. Our method is to employ a known class of ordinary convex polyhedra naturally associated with weighted digraphs. This way we can relate to and use results from combinatorics and optimization. One outcome is the solution of a conjecture of Develin and Yu (2007). We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the Abel-Jacobi, Riemann-Roch and Riemann theta divisor theorems. Abstract A new probabilistic clustering method, based on a regression mixture model, is used to describe tropical cyclone (TC) propagation in the western North Pacific (WNP). Seven clusters were obtained and described in Part I of this two-part study. In Part II, the present paper, the large-scale patterns of atmospheric circulation and sea surface temperature associated with each of the clusters are investigated, as well as associations with the phase of the El Nino–Southern Oscillation (ENSO). Composite wind field maps over the WNP provide a physically consistent picture of each TC type, and of its seasonality. Anomalous vorticity and outgoing longwave radiation indicate changes in the monsoon trough associated with different types of TC genesis and trajectory. The steering winds at 500 hPa are more zonal in the straight-moving clusters, with larger meridional components in the recurving ones. Higher values of vertical wind shear in the midlatitudes also accompany the straight-moving tracks, compared to...