  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.
