A new sparse octanomial normal form for cubic surfaces is defined, and p-adic tropical smoothness in this family forces the 27 lines to be distinct and defined over Q_p.
Computing zero-dimensional tropical varieties via projections
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abstract
We present an algorithm for computing zero-dimensional tropical varieties using projections. Our main tools are fast unimodular transforms of lexicographical Gr\"obner bases. We prove that our algorithm requires only a polynomial number of arithmetic operations if given a Gr\"obner basis, and we demonstrate that our implementation compares favourably to other existing implementations. Applying it to the computation of general positive-dimensional tropical varieties, we argue that the complexity for calculating tropical links is dominated by the complexity of the Gr\"obner walk.
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An Octanomial Model for Cubic Surfaces
A new sparse octanomial normal form for cubic surfaces is defined, and p-adic tropical smoothness in this family forces the 27 lines to be distinct and defined over Q_p.