Signed tropicalization of polars and application to matrix cones

We study the tropical analogue of the notion of polar of a cone, working over the semiring of tropical numbers with signs and show that the tropical polars of sets of nonnegative tropical vectors are precisely sets of signed vectors that are closed and that are stable by an operation of linear combination. We 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 a field, 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. It turns out, in particular, that hierarchies of classical cones collapse under tropicalization. We finally discuss a simple application of these ideas to optimization with signed tropical numbers.