Plucker-Type Inequalities for Mixed Areas and Intersection Numbers of Curve Arrangements

Document Type

Article

Publication Date

9-2023

Publication Title

International Mathematics Research Notices

Abstract

Any collection of n compact convex planar sets K-1,..., K-n defines a vector of ((n)(2)) mixed areas V(K-i, K-j) for 1 <= i < j <= n. We show that for n >= 4 these numbers satisfy certain Plucker-type inequalities. Moreover, we prove that for n = 4, these inequalities completely describe the space of all mixed area vectors (V(K-i, K-j) : 1 <= i < j <= 4). For arbitrary n >= 4, we show that this space has a semialgebraic closure of full dimension. As an application, we show that the pairwise intersection numbers of any collection of n tropical curves satisfy the Plucker-type inequalities. Moreover, in the case of four tropical curves, any homogeneous polynomial relation between their six intersection numbers follows from the corresponding Plucker-type inequalities.

DOI

10.1093/imrn/rnac216

Volume

2023

Issue

18

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