Basecondary polytopes

Abstract

Many (if not most) of convex polytopes, important for combinatorial and algebraic geometry, are closely related to secondary polytopes of point configurations, or base polytopes of submodular functions, or their numerous variations and generalizations. The aim of this text is to introduce the class of basecondary polytopes. This class includes (and allows to study uniformly) the aforementioned ones, as well as some others, e.g. appearing as Newton polytopes of important discriminant hypersurfaces. Most notably, this includes the discriminant of the Lyashko--Looijenga map, which is important for enumerative geometry of ramified coverings and cannot be reduced (by far) to Gelfand--Kapranov--Zelevinsky's A-discriminants and secondary polytopes.

Figures (6)

• Figure 1: A tropical polynomial with coinciding critical values (on the left) and a tropical polynomial with a degenerate critical point (on the right).
• Figure 2: The map $L\colon x\mapsto 1\cdot x$ is a generic $\gamma_1$-simplicial map.
• Figure 3: The map $L\colon x\mapsto 0\cdot x$ is $\gamma_2$-simplicial, but not generic.
• Figure 4: The map $L\colon x\mapsto 0\cdot x$ is $\gamma_3$-circuital.
• Figure 5: Circuits in dimension 2.

Theorems & Definitions (2)

• proof
• proof : Proof of Lemma \ref{['lemma:iterated']}