Tropical Ideals and Discriminants

Tropical ideals arose in the work of Maclagan-Rincon and Giansiracusa-Giansiracusa in the context of their scheme-theoretic renement of tropicalization. An open problem is to understand what geometric information about a variety V (I) is encoded in the tropical ideal trop(I). In this thesis we focus on the valuation of the discriminant for certain classes of projective hypersurfaces of low-degree in both characteristic 0 and p. We nd both cases where the valuation of the discriminant is determined by trop(I), and cases where it is not.