# Signed Tropicalization of Polar Cones

## TL;DR

This paper explores the tropical version of polar cones and how they relate to classical cones through nonarchimedean valuation, with applications in optimization.

## Contribution

The paper introduces a signed tropicalization framework for polar cones and shows how classical cone hierarchies collapse under tropicalization.

## Key findings

- Tropical polars are characterized by an invariance property under tropical Fourier–Motzkin elimination.
- Signed valuation commutes with taking the polar for semi-algebraic sets over nonarchimedean fields.
- Tropicalization causes hierarchies of classical cones, such as positive semidefinite and co-positive matrices, to collapse.

## Abstract

We study the tropical analogue of the notion of polar of a cone, working over the semiring of tropical numbers with signs. We characterize the cones which arise as polars of sets of tropically nonnegative vectors by an invariance property with respect to a tropical analogue of Fourier–Motzkin elimination. We also relate tropical polars with images by the nonarchimedean valuation of classical polars over real closed nonarchimedean fields and show, in particular, that for semi-algebraic sets over such fields, the operation of taking the polar commutes with the operation of signed valuation (keeping track both of the nonarchimedean valuation and sign). We apply these results to characterize images by the signed valuation of classical cones of matrices, including the cones of positive semidefinite matrices, completely positive matrices, completely positive semidefinite matrices, and their polars, including the cone of co-positive matrices, showing that hierarchies of classical cones collapse under tropicalization. We finally discuss an application of these ideas to optimization with signed tropical numbers.