Extremal decompositions of tropical varieties and relations with rigidity theory

Abstract

Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the extremal decomposition of tropical varieties with rational balanced weightings. Additionally, we explore connections and applications related to rigidity theory. In particular, we prove that a tropical hypersurface is extremal if and only if it has a unique reciprocal diagram up to homothety.

Figures (12)

• Figure 1: Examples of tropical varieties from \ref{['Ex:tropical-varieties']}
• Figure 2: The tropical variety $\mathbb{V}(0 \oplus x \oplus y \oplus z)$
• Figure 3: A weighted tropical variety which can not be decomposed.
• Figure 4: The tropical variety from \ref{['fig:Red-Trop-Var-Bad-W']} with weighting scaled by 3 and the corresponding decomposition.
• Figure 5: An extremal tropical variety

Theorems & Definitions (59)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Example 2.2
• Definition 2.3
• Proposition 2.4
• Remark 2.5
• proof : Proof of \ref{['lem:extre-decomp']}