Signed Tropicalization of Polar Cones

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.