Signed Tropicalization of Polar Cones

Area of Science:

• Algebraic Geometry
• Optimization Theory
• Tropical Algebra

Background:

• The study of polars of cones is fundamental in convex geometry and optimization.
• Tropical algebra provides a framework for analyzing algebraic structures over specific semirings.

Purpose of the Study:

• To define and characterize the tropical analogue of the polar of a cone over the semiring of tropical numbers with signs.
• To investigate the relationship between tropical polars and classical polars using nonarchimedean valuations.
• To apply these tropical concepts to analyze classical matrix cones and their hierarchies.

Main Methods:

• Utilizing the semiring of tropical numbers with signs.
• Employing a tropical analogue of Fourier-Motzkin elimination for cone characterization.
• Relating tropical polars to classical polars via nonarchimedean valuation and signed valuation.
• Analyzing semi-algebraic sets over real closed nonarchimedean fields.

Main Results:

• Characterization of tropical polars through an invariance property under tropical Fourier-Motzkin elimination.
• Demonstration that the polar operation commutes with signed valuation for semi-algebraic sets.
• Identification of the collapse of hierarchies of classical matrix cones (e.g., positive semidefinite, completely positive) under tropicalization.
• Characterization of images of classical cones and their polars (e.g., co-positive matrices) under signed valuation.

Conclusions:

• The tropical polar provides a powerful tool for understanding geometric and algebraic structures in tropical algebra.
• The commutation of polar and signed valuation simplifies analysis of matrix cones over nonarchimedean fields.
• Tropicalization offers a novel perspective on matrix cone structures and their optimization properties.