The Signed Goldman-Iwahori Space and Real Tropical Linear Spaces

Abstract

For a real closed field $K$ with a non-Archimedean absolute value, we introduce the signed Goldman--Iwahori space, the space of homothety classes of signed seminorms on a finite-dimensional vector space over $K$. This space merges two geometric perspectives: it is the linear-algebraic analogue of the real analytification of projective space introduced by Jell, Scheiderer, and Yu, and a signed refinement of the Goldman--Iwahori space of seminorms studied in our previous work. Our first main result identifies it as the inverse limit of all real tropicalizations of projective space, the real analogue of a theorem of Payne. Our second main result gives a matroid-theoretic description in terms of the universal realizable oriented valuated matroid on the underlying vector space. In the constant coefficient case for $K = \mathbb{R}$, signed seminorms are exactly the diagonalizable ones which is a consequence of hyperplane separation that fails for all other real closed fields. In this case, this yields an explicit description of the signed Goldman--Iwahori space in terms of signed flags of subspaces and lets us show that the space coincides with the real Bergman fan of the universal oriented matroid.