Combinatorial or polyhedral description for tropicalization of the positive subset of a real linear subspace

Question

I had two questions: one regarding a definition of tropicalized linear subspaces, and the second about how to find similar characterizations for the logarithmic limit set of the positive subset of a linear subspace.

I've been reading Maclagan and Sturmfel's "Introduction to Tropical Geometry", and in Chapter 4.1 and 4.2, they discuss the tropicalization of linear spaces. For a valued field $K$, given $n+1$ column vectors $\textbf{b}_i \in K^{d+1}$, for $n>d$, they define a matrix $\textbf{B}$ whose columns are the $\textbf{b}_i$, and later on define a tropicalized linear space associated to $\textbf{B}$ via the tropical basis formed by the linear polynomials whose supports correspond to circuits of $\textbf{B}$. In Chapter 4.2, they also characterize $trop(L)$ in terms of $\textbf{B}$'s associated matroid's flats and polyhedra.

While the book defines a hyperplane arrangement associated to $\textbf{B}$, I didn't intuitively understand its connection to the eventual tropical linear space. If I were to reinterpret, I feel like the rows of $\textbf{B}$ span a $d+1$-linear subspace $L \subset K^{n+1}$, and the tropicalized linear space defined by $\textbf{B}$ in the book is the tropicalization of $L$. So first, if this is incorrect, how would spanning vectors of a linear subspace $L$, relate to its tropicalization as a variety?

My main question: following the perspective that the tropicalization of a variety, as a set, corresponds to taking the logarithmic limit of that set (via Maslov dequantization), take the intersection of a real linear subspace $L^+ \equiv L \cap \mathbb{R}_{>0}^n $, and its logarithmic limit set $trop(L^+)$. What are the combinatorial (e.g. linear forms related to circuits) or polyhedral (e.g. the flats of the matroid) descriptions of the support of $trop(L^+)$? Where should I look to get a better understanding? My endgoal is to describe $trop(L^+)$ given a spanning set of $L$.

While I don't fully understood the papers in the field, I was looking for an answer in:

"Logarithmic limit sets of real semi-algebraic sets" https://arxiv.org/abs/0707.0845

"Tropical Linear Spaces and Tropical Convexity" https://arxiv.org/abs/1505.02045

"The tropical totally positive Grassmannian" https://arxiv.org/abs/math/0312297

Thanks for any help or clarification.

Answer 1

Thanks for the help Joshua.

I've found a paper that addresses my original question.

https://arxiv.org/pdf/math/0406116.pdf

The key difference between the standard tropical linear space and the positive tropical linear space is that one moves from matroids to oriented matroids.

Answer 2

I can't answer your question about logarithmic limits, but I can answer the one about tropicalizations. (It might be useful to make two separate questions.)

Maclagan and Sturmfels' construction indeed agrees with the tropicalization of the row space of $\mathbf{B}$. Their construction in terms of circuits gives a tropical basis for the ideal $I$ of the (projectivized) row space of $\mathbf{B}$ (in the torus). This corresponds to taking the tropicalization of the kernel of $\mathbf{B}$, then taking the tropical orthogonal complement.

If we set $Y = ker(\mathbf{B})$ and $X = im(\mathbf{B}^T)$, then $X$ and $Y$ are orthogonal complements in $K^{n+1}$ (via the standard dot product). Maclagan and Sturmfels' construction results in $trop(Y)^\perp$, where here we mean the tropical orthogonal complement $$ trop(Y)^\perp = \{x \in \mathbb{R}^{n+1}/\mathbb{R}\mathbf{1} : \text{ the minimum of } x_i+y_i \text { is achieved twice}\}.$$ But this is equal to $trop(X) = trop(Y^\perp)$, in the sense of taking coordinate-wise valuation of entries of elements of $X$. Using only techniques of tropical linear spaces, perhaps the easiest way to see this is through tropical Pluecker coordinates, as the formula for the Pluecker coordinates of the classical orthogonal dual tropicalize to the formula for the tropical Pluecker coordinates of the tropical orthogonal dual.

This is a manifestation of the Fundamental Theorem of Tropical Geometry, which states that for an ideal $I$ of Laurent polynomials over $K$, coordinate-wise valuation of $V(I)$ is equal to the tropical variety defined by $trop(I)$.