Introduction↩︎

The Goldman–Iwahori space is the space of non-Archimedean seminorms on a finite-dimensional vector space over a non-Archimedean field. It was introduced in [1] to give a non-Archimedean analogue of symmetric spaces for the group \(\mathop{\mathrm{PGL}}\). If the ground field is spherically complete, this space is a compactification of a building in the sense of Bruhat and Tits. Bruhat–Tits buildings have proven to be an effective framework to study reductive groups over non-Archimedean fields (see [2], [3]). Their close relation to Berkovich geometry is well-established in the literature (see [4]–[7]). Expanding on [8], the authors in [9] introduced a tropical approach to study the Goldman–Iwahori space by identifying it with the limit of tropicalized linear spaces and giving a matroidal interpretation.

Suppose the ground field is now also real closed such that the unique ordering is compatible with the absolute value. This paper presents a signed analogue of the Goldman–Iwahori space, taking the order of the field into consideration. We approach this signed Goldman–Iwahori space by methods of real tropical geometry. This extends the non-Archimedean framework to real tropical geometry, which was introduced in [10].

We recall some basics of tropical geometry: let \(K\) be an algebraically closed field with a non-trivial non-Archimedean absolute value \(\lvert\cdot \rvert\). We denote the tropicalization map by \[\mathop{\mathrm{trop}}: K^n \to (\mathbb{R}\cup \{\infty\})^n,\quad (x_1,\dots,x_n) \mapsto (-\log |x_1|,\dots,-\log|x_n|)\, .\] The closure of the image of an algebraic subvariety \(X\subseteq K^n\) under \(\mathop{\mathrm{trop}}\) is called the tropicalization of \(X\) and usually denoted by \(\mathop{\mathrm{Trop}}(X)\). This space has the structure of a finite polyhedral complex of dimension \(\dim X\) and its combinatorics carries rich information about the variety \(X\), like its degree and Chow cohomology class.

It is a fundamental trait of tropical geometry that \(\mathop{\mathrm{Trop}}(X)\) not only depends on the variety \(X\), but also on the chosen embedding into \(K^n\). Hence, we will write the tropicalization as \(\mathop{\mathrm{Trop}}(X,\iota)\) to emphasize the dependence on the embedding \(\iota: X\hookrightarrow K^n\). More generally, one can define tropicalizations of closed subvarieties of any toric variety, not just \(K^n\) [11]. One can also drop the conditions of \(K\) being algebraically closed or non-trivially valued by choosing an algebraically closed, non-trivially field valued extension \(L/K\) and defining \(\mathop{\mathrm{Trop}}(X,\iota) \mathrel{\vcenter{:}}= \mathop{\mathrm{Trop}}(X \times_K L, \iota_L)\).

A natural question is, whether there is a universal tropicalization of a variety that is independent of the embedding. One way to construct this, is taking the projective limit of all tropicalizations with respect to all possible embeddings for a quasi-projective variety \(X\). In the influential paper [11], Payne showed that this limit space is homeomorphic to the Berkovich analytification \(X^{\mathop{\mathrm{an}}}\) of \(X\). This analytic space \(X^{\mathop{\mathrm{an}}}\), defined by Berkovich, is a connected Hausdorff topological space that contains the set of \(K\)-points \(X(K)\) as a dense subset (see [4] for details). Payne’s result led to ample research into limits of tropicalizations, for example in [12]–[14].

Recently, there have been two new developments that this paper combines:

(a) If one considers \(X=\mathbb{P}^n\), in [9] it was shown that the Goldman–Iwahori space \(\overline{\mathcal{X}}_n(K)\) is the limit of all tropicalizations with respect to linear embeddings of \(\mathbb{P}^n\) into higher-dimensional projective spaces. This space \(\overline{\mathcal{X}}_n(K)\) consists of homothety classes of seminorms on \((K^{n+1})^{\ast}\) and has first been studied by Goldman and Iwahori as a piecewise linear analogue of symmetric spaces [1] for the group \(\mathop{\mathrm{PGL}}(K^{n+1})\). If \(K\) is spherically complete, it coincides with (a compactification of) the affine Bruhat–Tits building of \(\mathop{\mathrm{PGL}}(K^{n+1})\), e.g.studied by Werner in [5]. The relation between affine buildings and Berkovich analytic spaces has been well established in [4], [6], [7], [15].

(b) In [10], the authors define a “real” version of a Berkovich analytification: In this setup the field \(K\) is real closed instead of algebraically closed and a real analytification \(X_r^{\mathop{\mathrm{an}}}\) of a variety \(X\) is constructed, which takes into account the order on \(K\). The authors introduce a non-Archimedean approach to real tropicalizations using the real analytification. Roughly speaking, given an affine variety \(X\) over \(K\), one restricts the tropicalization map to each orthant of \(K^n\) and glues the resulting \(2^n\) tropicalizations together. The resulting space \(\mathop{\mathrm{Trop}}_r(X, \iota)\) is called the real tropicalization of \(X\) with respect to the embedding \(\iota:X \xhookrightarrow{} K^n\). In [10], it was shown that \(X_r^{\mathop{\mathrm{an}}}\) is the limit of all real tropicalizations. Real tropicalizations have garnered particular attention due to Viro’s patchworking [16], which was one of the earliest achievements of what is now called tropical geometry. An equivalent perspective on real tropicalizations is given by real phase structures, and recently, in [17] the authors defined the real part of a smooth tropical variety equipped with a real phase structure, which is locally (the topological realization of) an oriented matroid [18].

We merge these two approaches: Consider a real closed field \(K\) equipped with a compatible non-Archimedean absolute value. An example is the field of real Puiseux series \(\mathbb{R}\{\!\{t\}\!\} =\bigcup_{n \in \mathbb{N}} \mathbb{R}(\!(t^{\frac{1}{n}})\!)\), where the positive Puiseux series are those with positive leading coefficients. Let \(X=\mathbb{P}^n\) be the projective space over \(K\). We define the signed Goldman–Iwahori space \(\mathfrak{R}\mathcal{X}_n(K)\) to be the space of homothety classes of non-trivial signed seminorms on \((K^{n+1})^{\ast}\) (cf. Definition 7) equipped with the topology of pointwise convergence. We study the geometry of this space via real tropical geometry, in particular via real tropical linear spaces:

Let \(I\) be the cofiltered category of linear embeddings \(\mathbb{P}^n \hookrightarrow U \subseteq \mathbb{P}^m\), where \(U\) is a torus-invariant open subset of \(\mathbb{P}^m\) with morphisms given by commutative triangles

where \(U\to U'\) is a composition of coordinate projections and permutations. For any linear embedding \(\iota: \mathbb{P}^n \hookrightarrow U\subseteq \mathbb{P}^m\) as above, we construct a surjective and continuous map \(\pi_{\iota}: \mathfrak{R}\mathcal{X}_n(K)\to \mathop{\mathrm{Trop}}_r(\mathbb{P}^n,\iota)\). For any morphism in \(I\), the coordinate projections and permutations induce a continuous map between the respective real tropicalizations. This setup allows us to formulate and prove:

Theorem 1 can both be viewed as a signed version of [9] or as a linear version of [10]. Even though the signed Goldman–Iwahori space \(\mathfrak{R}\mathcal{X}_n(K)\) is not a building in the sense of Bruhat and Tits, it can be seen as a signed analogue of the building of \(\mathop{\mathrm{PGL}}\). From this perspective, it is a real analytic and non-Archimedean analogue of a symmetric space for \(\mathop{\mathrm{PGL}}\). It might be interesting to examine this space more closely in order to study reductive groups over real closed fields. Taking absolute values yields a natural, surjective, and continuous map \(\mathfrak{R}\mathcal{X}_n(K)\to \overline{\mathcal{X}}_n(K)\) to the Goldman–Iwahori space (cf. Section 5.1).

Valuated matroids are the combinatorial objects that naturally arise with (ordinary) tropicalizations of linear spaces. To every linear embedding \(\iota: \mathbb{P}^n \hookrightarrow \mathbb{P}^m\) we may associate a valuated matroid that uniquely determines the tropicalization \(\mathop{\mathrm{Trop}}(\mathbb{P}^n,\iota)\). Even in the non-realizable case, one can associate a linear space to a valuated matroid, which agrees with the Bergman fan in the case of trivial valuation. In [9], the construction was generalized to allow us to describe finite rank matroids on infinite ground sets. In [9], it was also shown, that the Goldman–Iwahori space \(\overline{\mathcal{X}}_n(K)\) is the tropical linear space associated to the universal realizable valuated matroid \(M_{\mathop{\mathrm{univ}}}\). This matroid has as ground set \((K^{n+1})^{\ast}\) and the rank function is given by the dimension of the subspace spanned by a subset.

When the field \(K\) is real closed (or, more generally, ordered), the additional decoration on a matroid is that of an orientation, which has been studied extensively [19], [20]. As matroids capture the combinatorics of linear dependence over fields, oriented matroids capture the combinatorics of linear dependence over ordered fields, taking into account signs of linear dependencies. If the field \(K\) is in addition equipped with a compatible non-Archimedean value, we obtain an oriented valuated matroid, which is a matroid with both decorations that satisfy a compatibility condition.

Baker and Bowler introduce matroids with decorations in the language of hyperfields [21], where valuated/oriented/oriented valuated matroids arise as matroids over the tropical hyperfield \(\mathbb{T}\)/the sign hyperfield \(\mathbb{S}\)/the real tropical hyperfield \(\mathbb{R}\mathbb{T}\).

To every oriented valuated matroid \(\mathcal{M}\) on a ground set \(E\), one can associate a real tropical linear space \(\mathop{\mathrm{Trop}}_r(\mathcal{M})\) [22], [23]. If \(\mathcal{M}\) is represented by a linear embedding over a real closed field, then the real tropicalization of the linear embedding coincides with the real tropical linear space associated to \(\mathcal{M}\). This extends the trivial valuation case considered in [24], [25] who shows that the real tropical linear space equals the real Bergman fan of the associated oriented matroid.

We generalize these ideas and construct a real tropical linear space for any oriented valuated matroid of finite rank, whose ground set explicitly need not be finite (see Section 6). We then show:

Here, \(\mathcal{M}_{\mathop{\mathrm{univ}}}\) denotes the universal realizable oriented valuated matroid with ground set \(E=(K^{n+1})^{\ast}\). This gives \(\mathfrak{R}\mathcal{X}_n(K)\) an appealing interpretation as the real tropicalization of the natural universal embedding \[\iota_{\mathop{\mathrm{univ}}}: \mathbb{P}^{n}\longhookrightarrow \mathbb{P}\left( K^{E}\right) \, .\] In particular, this allows for a notion of a real tropicalization of an embedding of a finite-dimensional variety into an infinite-dimensional vector space. This gives an interpretation of the signed Goldman–Iwahori space as a universal realizable real tropical linear space.

In the special case \(K=\mathbb{R}\) with trivial valuation signed seminorms are always diagonalizable, which need not even be true for other trivially valued fields (cf.Example 10). Therefore, in general, computing signed Goldman–Iwahori spaces turns out to be difficult. However, if \(K=\mathbb{R}\), we can explicitly parameterize the signed Goldman–Iwahori space and determine the fibers of the map \(\mathfrak{R}\mathcal{X}_n(\mathbb{R})\to \overline{\mathcal{X}}_n(\mathbb{R})\).

In the case of trivial valuation, the real tropical linear space of an oriented matroid agrees with its real Bergman fan studied in [25]. We show that for the universal realizable oriented matroid, this is still true if \(K=\mathbb{R}\), but fails for other real closed fields (cf. Theorem 37 and Remark 38 (d)). Tropical linear spaces relate to tropical convexity (see [26] for details), in the same way that real tropical linear spaces relate to signed tropical convexity [27]. The real tropicalization of a linear space is the TC-convex hull of all \(\mathbb{R}\mathbb{T}\)-cocircuits of the associated \(\mathbb{R}\mathbb{T}\)-matroid [27]. We give an interpretation of this result for the real tropicalization of the universal realizable oriented matroid \(\mathcal{M}_{\mathop{\mathrm{univ}}}\).

Conventions↩︎

We write \(\overline{\mathbb{R}} = \mathbb{R}\cup \{\infty\}\). For any set \(E\), we write \(\mathbb{T}\mathbb{P}^E \mathrel{\vcenter{:}}= \big(\overline{\mathbb{R}}^E\setminus \infty\big)/\mathbb{R}\mathbf{1}\), where \(\mathbf{1}\) refers to the function with constant value \(1\). If \(E=\{0,\dots,n\}\) for some natural number \(n\), we write \(\mathbb{T}\mathbb{P}^n\) for \(\mathbb{T}\mathbb{P}^E\). For a field \(K\), which will be clear from the context, we write \(\mathbb{A}^n = \mathop{\mathrm{Spec}}(K[t_1,\dots,t_n])\) for affine space and \(\mathbb{P}^n = \mathop{\mathrm{Proj}}( K[t_0,\dots,t_n])\) for projective space. If \(K=\mathbb{R}\), we write \(\mathbb{R}\mathbb{P}^n\) for the set of \(\mathbb{R}\)-points of \(\mathbb{P}^n\) and consider this space with the Euclidean topology. We denote the field with two elements by \(\mathbb{Z}_2 = \{0,1\}\). We consider the ordering \(+,- > 0\) of signs which we extend componentwise to a partial order on sign vectors. Given an ordered field \(K\), we denote by \(\mathop{\mathrm{sgn}}: K \to \{0,+1,-1\}\) the sign function.

Acknowledgements↩︎

This project was initiated 2023 at the Graduate Student Meeting on Applied Algebra and Combinatorics in Stockholm, where we had the opportunity to present our project on buildings and tropical linear spaces [9]. Kris Shaw pointed us to the real analytification and real tropicalization of Jell, Scheiderer, and Yu, thus inspiring us to investigate a real/signed version of our story. We thank Kris Shaw and further all speakers and organizers at this occasion. Additionally, the authors want to thank Luca Battistella, Andreas Gross, Alex Küronya, Georg Loho, Oliver Lorscheid, Alejandro Martinez Mendez, Beatrice Pozzetti, Kemal Rose, Ben Smith, Pedro Souza, Martin Ulirsch, Alejandro Vargas, and Annette Werner for helpful conversations and discussions at various stages of the project.

Funding↩︎

This project has received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and Arithmetic of Uniformized Structures, project number 444845124, from the DFG Sachbeihilfe From Riemann surfaces to tropical curves (and back again), project number 456557832, as well as the DFG Sachbeihilfe Rethinking tropical linear algebra: Buildings, bimatroids, and applications, project number 539867663, within the SPP 2458 Combinatorial Synergies.

1 Real Tropical Geometry↩︎

We describe the real part of tropical projective space which is homeomorphic to the real projective space \(\mathbb{R}\mathbb{P}^n\). This space is the ambient space of the real tropicalizations of subvarieties of \(\mathbb{P}_K^n\) over a real closed field \(K\) with compatible absolute value. We describe this construction of a real tropicalization, which, in contrast to ordinary tropicalization, also takes the order on \(K\) into account. A more general and thorough treatment can be found for example in [17].

1.1 Tropical Patchworking↩︎

Let \(\overline{\mathbb{R}} = \mathbb{R}\cup \{\infty\}\). For \(n\in \mathbb{N}\), the tropical projective space of dimension \(n\) is given by \(\mathbb{T}\mathbb{P}^n\mathrel{\vcenter{:}}= \overline{\mathbb{R}}^{n+1} \setminus \{(\infty, \dots, \infty )\} /\mathbb{R}\mathbf{1}\), where \(\mathbf{1}=(1,\dots,1)\). Via the identification \(\overline{\mathbb{R}}\cong \mathbb{R}_{\geq 0},\;x \mapsto \exp(-x)\), we can identify \(\mathbb{T}\mathbb{P}^n \cong \mathbb{R}_{\geq 0}^{n+1} \setminus \{(0,\dots,0)\} / ( \mathbb{R}_{>0})\). The space \(\mathbb{T}\mathbb{P}^n\) is naturally stratified, where the closures of strata are given by \(\{(x_0:\dots:x_n)\in \mathbb{T}\mathbb{P}^n\mid x_i=\infty\; \forall i \in I\}\), where \(I\subsetneq \{0,\dots,n\}\).

We can glue together \(2^n\) symmetric copies of \(\mathbb{T}\mathbb{P}^n\) indexed by \(\varepsilon\in \mathbb{Z}_2^{n+1}/\mathbf{1}\) to obtain a space \[\mathfrak{R}\mathbb{T}\mathbb{P}^n \mathrel{\vcenter{:}}= \bigcup_{\varepsilon \in \mathbb{Z}_2^{n+1}/\mathbf{1}} \mathbb{T}\mathbb{P}^n(\varepsilon) /\sim.\] Here, \(\sim\) identifies the codimension-one strata \(\{(x_0:\dots: x_n)\in \mathbb{T}\mathbb{P}^n(\varepsilon) \mid x_i=\infty\}\) and \(\{(x_0:\dots: x_n)\in \mathbb{T}\mathbb{P}^n(\varepsilon') \mid x_i=\infty\}\) if \(\varepsilon+\varepsilon'=e_i\) in \(\mathbb{Z}_2^{n+1}/\mathbf{1}\). The identification \(\mathbb{T}\mathbb{P}^n \cong \mathbb{R}_{\geq 0}^{n+1} \setminus \{(0,\dots,0)\} / ( \mathbb{R}_{>0})\) yields an identification \[\begin{align} \mathfrak{R}\mathbb{T}\mathbb{P}^n &\cong \mathbb{R}\mathbb{P}^n, \\ (\varepsilon, (x_0:\dots:x_n)) &\mapsto \bigl((-1)^{\varepsilon_0}\exp(-x_0):\dots:(-1)^{\varepsilon_n}\exp(-x_n)\bigr). \end{align}\]

There is an obvious continuous retraction map \(\lvert \cdot \rvert_{\mathbb{R}\mathbb{P}^n}: \mathbb{R}\mathbb{P}^n \to \mathbb{R}_{\geq 0}^{n+1} \setminus \{(0,\dots,0)\} / ( \mathbb{R}_{>0}) \cong \mathbb{T}\mathbb{P}^n\), sending each \((x_0:\dots : x_n) \in \mathbb{R}\mathbb{P}^n\) to \((|x_0|:\dots:|x_n|)\), where \(\lvert \cdot \rvert\) denotes the absolute value on the real numbers.

Example 1. We make the space \(\mathfrak{R}\mathbb{T}\mathbb{P}^2 \cong \mathbb{R}\mathbb{P}^2\) explicit by gluing four triangles, each homeomorphic to \(\mathbb{T}\mathbb{P}^2\), together as in Figure 1 and identifying antipodal points. The retraction map \(\lvert \cdot \rvert_{\mathbb{\mathbb{R}}\mathbb{P}^2}\) folds up the four triangles to one.

For the sake of clarity, we will henceforth always use the coordinates of \(\mathbb{R}\mathbb{P}^n\) and omit any logarithms. The downside is, that our tropicalizations are not polyhedral complexes in these coordinates. However, the formulas are much clearer since these coordinates make it easier to keep track of the signs.

1.2 Ordered and Real Closed Fields↩︎

We recall the definition of an ordering on a ring: For a commutative ring \(A\), a subset \(P \subseteq A\) is called an ordering of \(A\) if \(P+P \subseteq P, P \cdot P \subseteq P, P \cup -P = A,\) and \(P \cap -P\) is a prime ideal of \(A\). The support of \(P\) is \(\text{supp}(P) \mathrel{\vcenter{:}}= P \cap -P\). To \(P\) we associate an order relation and a sign function in the natural way: Write \(f >_P 0\) and \(\mathop{\mathrm{sgn}}_P(f) = +1\) if \(-f \notin P\), write \(f \geq_P 0\) if \(f \in P\), and \(\mathop{\mathrm{sgn}}_P(f) = 0\) if \(f \in \text{supp}(P)\). We have \(\mathop{\mathrm{sgn}}_P(-f) = - \mathop{\mathrm{sgn}}_P(f)\). We may refer to the ordering as either \(P\) or \(<_P\). An ordered field \(K\) is real closed if every non-negative element \(x \geq 0\) has a square root in \(K\). If \(K\) is real closed, there is a unique ordering on \(K\) defined by taking the positive elements to be precisely the non-zero squares. Every ordered field has a real closed extension, called the real closure. Whenever a real closed field is equipped with a non-Archimedean absolute value \(\lvert \cdot \rvert_K\), we assume that the absolute value is compatible with the (unique) ordering. This means, if \(0 \leq a \leq b\) then \(|a|_K \leq |b|_K\), or equivalently, if \(|a|_K > |b|_K\) then \(\mathop{\mathrm{sgn}}(a+b) = \mathop{\mathrm{sgn}}(a) = \mathop{\mathrm{sgn}}(a-b)\).

1.3 Real Tropicalization↩︎

We will begin by providing a brief overview of ordinary tropicalization. Let \(K\) be an algebraically closed field with a non-trivial non-Archimedean absolute value \(\lvert \cdot \rvert_K: K\to \mathbb{R}_{\geq 0}\). Denote the tropicalization map by \[\begin{align} \mathop{\mathrm{trop}}: \mathbb{P}^n(K) &\longrightarrow (\mathbb{R}_{\geq 0}^{n+1} \setminus \{0\}) / \mathbb{R}_{>0}\, ,\\ (x_0: \dots :x_n) &\longmapsto (|x_0|_K : \dots : |x_n|_K). \end{align}\] For any closed subvariety \(\iota: X\hookrightarrow \mathbb{P}^n\) the tropicalization \(\mathop{\mathrm{Trop}}(X,\iota)\) is defined to be the closure of the image of \(X\) under \(\mathop{\mathrm{trop}}\). If \(K\) is trivially valued or not algebraically closed, we pass to a non-trivially valued, algebraically closed extension \(L/K\) and define \(\mathop{\mathrm{Trop}}(X,\iota)\mathrel{\vcenter{:}}= \mathop{\mathrm{Trop}}(X_L,\iota_L) \subseteq (\mathbb{R}_{\geq 0}^{n+1} \setminus \{0\}) / \mathbb{R}_{>0}\), where \(X_L= X\times_K L\). This is independent of the choice of \(L\) (e.g.by [29]).

Let now \(K\) be real closed instead of algebraically closed. We assume that \(\lvert \cdot \rvert_K\) is compatible with the unique ordering on \(K\). The real tropicalization map is given by \[\begin{align} \mathop{\mathrm{trop}}_r: \mathbb{P}^n(K) &\longrightarrow \mathbb{R}\mathbb{P}^n\, , \\ (x_0: \dots :x_n) &\longmapsto (\mathop{\mathrm{sgn}}(x_0) |x_0|_K : \dots :\mathop{\mathrm{sgn}}(x_n) |x_n|_K). \end{align}\] Note that this map takes into account the order on \(K\). This can be seen as an orthantwise tropicalization, where we restrict the tropicalization map to each orthant and glue the resulting tropicalizations together. The affine version of this map is considered by Jell, Scheiderer, and Yu in [10] and [32], where the authors study properties of the images of semialgebraic sets under the real tropicalization map. The logarithmic version of this construction, without signs, was used by Alessandrini in [33] who showed that the logarithmic limit of a real semialgebraic set is a polyhedral complex.

As before, for every closed subvariety (or, more generally, for every semialgebraic subset) \(\iota: X\hookrightarrow \mathbb{P}^n\), we define the real tropicalization \(\mathop{\mathrm{Trop}}_r(X, \iota) \subseteq \mathbb{R}\mathbb{P}^n\) as the closure of the image of \(X(K)\) under \(\mathop{\mathrm{trop}}_r\). If \(K\) is trivially valued, we choose a non-trivially valued, real closed field extension \(L/K\) such that the absolute value on \(L\) extends the one on \(K\) and define \(\mathop{\mathrm{Trop}}_r(X, \iota)\mathrel{\vcenter{:}}= \mathop{\mathrm{Trop}}_r(X_L, \iota_L)\subseteq \mathbb{R}\mathbb{P}^n\). The real tropicalization is independent of the choice of extension by [10]. By construction, the map \(\lvert \cdot \rvert_{\mathbb{R}\mathbb{P}^n}: \mathbb{R}\mathbb{P}^n \to \mathbb{T}\mathbb{P}^n\) maps \(\mathop{\mathrm{Trop}}_r(X,\iota)\) to \(\mathop{\mathrm{Trop}}(X,\iota)\).

Example 3. Let \(K=\mathbb{R}\) be equipped with the trivial valuation and let \(X=V(x_0+x_1-x_2)\subseteq \mathbb{P}^2_{\mathbb{R}}\). We want to compute \(\mathop{\mathrm{Trop}}_r(X,\iota)\), where \(\iota: X\hookrightarrow \mathbb{P}^2\) is the inclusion. Consider the base change to the field \(L\) of real Puiseux series. Let \((y_0:y_1:y_2) \in X_L(L)=V(x_0+x_1-x_2)\subseteq \mathbb{P}^2_L\), i.e., \(y_0+y_1=y_2\). Then \[\mathop{\mathrm{sgn}}((y_0,y_1,y_2))\in \{(+,+,+),(-,-,-),(+,-,-),(-,+,+),(+,-,+),(-,+,-)\},\] i.e., if \(\mathop{\mathrm{sgn}}(y_0)=\mathop{\mathrm{sgn}}(y_1)\), then all signs are the same. By the axioms of a non-Archimedean valuation it is clear that \(|y_2|\leq \max(|y_0|,|y_1|)\) and that, if \(|y_0|\neq |y_1|\), we have equality. Moreover, we also have equality if \(\mathop{\mathrm{sgn}}(y_0)=\mathop{\mathrm{sgn}}(y_1)\). Finally, if \(\mathop{\mathrm{sgn}}(y_0)\neq \mathop{\mathrm{sgn}}(y_1)\) and \(|y_0|= |y_1|\), then \(\mathop{\mathrm{sgn}}(y_2)\cdot|y_2|\) can have any value in \([-|y_0|,|y_0|]\cap \mathbb{Q}\). We may choose representatives in \(\mathbb{P}^2\) such that \(\max\{|y_0|,|y_1|,|y_2|\}=1\) and \(y_2\geq 0\). In Figure 2 the real tropicalization \(\mathop{\mathrm{Trop}}_r(X,\iota)\) is now indicated in red and the (usual) tropicalization \(\mathop{\mathrm{Trop}}(X,\iota)\) in blue.

2 Matroids over the Real Tropical Hyperfield↩︎

Matroids over hyperfields due to Baker and Bowler [21], [34] simultaneously generalize linear subspaces, matroids, oriented matroids, and valuated matroids. We recall their notion as well as the hyperfields that are of our interest such as the sign hyperfield, the tropical hyperfield, and, most importantly, the real tropical hyperfield. Matroids over the sign hyperfield correspond to oriented matroids, while matroids over the tropical hyperfield correspond to valuated matroids. Oriented valuated matroids are hybrid objects combining oriented matroids and valuated matroids in a compatible way. From the perspective of matroids over hyperfields they arise naturally as matroids over the real tropical hyperfield \(\mathbb{R}\mathbb{T}\). We recall the definition of a real tropical linear space associated to an oriented valuated matroid defined via its \(\mathbb{R}\mathbb{T}\)-circuits.

2.1 Hyperfields↩︎

In [35] Viro introduced hyperfields as a convenient technique in tropical geometry. Especially in the last few years there has been a surge of interest and research in the realm of hyperfields and tropical geometry [21], [36]–[39]. A hyperfield \(\mathbb{H}\) is a set with a multiplication \(\cdot\) and addition \(\oplus\), where addition may be multivalued, that satisfies axioms similar to those for a field. Several of the following hyperfields were first introduced in Viro’s paper [35], to which we refer for precise definitions.

A homomorphism of hyperfields is a map \(f: \mathbb{H}_1 \longrightarrow \mathbb{H}_2\) such that \(f(0) = 0, f(1) = 1, f(x \cdot y) = f(x) \cdot f(y)\), and \(f(x \oplus y) \subseteq f(x) \oplus f(y)\) for \(x,y \in \mathbb{H}_1\).

2.2 Matroids over Hyperfields↩︎

For the hyperfields of Example 4, we obtain the following (classical) notions:

(a) A matroid over a field \(K\) is the same thing as a linear subspace of \(K^E\) of rank \(n\). This is the classical representation of a subspace via its Plücker coordinates that satisfy the Grassmann–Plücker relations.

(b) When \(\mathbb{H}= \mathbb{K}\) is the Krasner hyperfield, a matroid over \(\mathbb{K}\) is the same as a usual matroid since the Grassmann–Plücker relations are equivalent to the basis exchange axiom for matroids.

(c) A matroid over the sign hyperfield \(\mathbb{S}\) is the same as an oriented matroid, where a choice of a Grassmann–Plücker \(\varphi\) function is called a chirotope (note that \(\varphi,-\varphi\) are the only chirotopes). This choice gives us then a notion of an oriented basis, where \((b_1,\dots,b_n)\) is a positively oriented basis, if \(\varphi(b_1,\dots,b_n)=1\).

(d) A matroid over the tropical hyperfield \(\mathbb{T}\) is the same as a valuated matroid in the sense of Dress–Wenzel [40]. The Grassmann–Plücker function is usually called a tropical Plücker vector.

(e) A matroid over the real tropical hyperfield \(\mathbb{R}\mathbb{T}\) is an oriented valuated matroid. This particular case holds significant interest within this paper.

Matroids over hyperfields admit a useful pushforward operation (cf.[21]): Given a Grassmann–Plücker function \(\varphi: E^n \xrightarrow{} \mathbb{H}_1\) and a homomorphism of hyperfields \(f: \mathbb{H}_1 \xrightarrow{} \mathbb{H}_2,\) the pushforward \(f_* \varphi: E^n \xrightarrow{} \mathbb{H}_2\) is defined by the formula \[\begin{align} f_*\varphi(e_1,\ldots,e_n) = f(\varphi(e_1,\ldots,e_n)). \end{align}\] This is again a Grassmann–Plücker function. If \(\mathcal{M}\) is a matroid over \(\mathbb{H}_1\) given by \(\varphi\), the \(\emph{pushforward}\) \(f_*(\mathcal{M})\) is defined by \(f_*\varphi\) and is a matroid over \(\mathbb{H}_2\).

2.3 Oriented Valuated Matroids↩︎

Since oriented valuated matroids are not as well studied as for example oriented or valuated matroids, we describe them now in more detail. The following explicit description is taken from [41]:

Let \(\mathcal{M}\) be an oriented valuated matroid given by the Grassmann–Plücker function \(\varphi: E^n \longrightarrow \mathbb{R}.\) Condition \((3)\) in Definition 1 explicitly means: For each \(\{x_1,\ldots,x_{n+1}\} \in E^{n+1}\) and \(\{y_1,\ldots,y_{n-1}\} \in E^{n-1}\), either the numbers \[\{(-1)^k \varphi(x_1,\ldots, \hat{x}_k,\ldots,x_{n+1}) \cdot \varphi(x_k,y_1,\ldots,y_{n-1}) \}_{k=1,\dots,n+1}\] are all zero, or the maximum modulus occurs with both signs. The group \(\mathbb{R}\mathbb{T}^* = \mathbb{R}^*\) acts on the set of such \(\varphi\) by multiplication, and an oriented valuated matroid is an orbit.

As in Example 6 (e) the pushforwards of an \(\mathbb{R}\mathbb{T}\)-matroid \(\mathcal{M}\) yield an underlying valuated matroid given by a tropical Plücker vector \(v=|\varphi|: E^{n} \longrightarrow \mathbb{T}= \mathbb{R}_{\geq 0}\) and an underlying oriented matroid given by the chirotope (i.e., Grassmann–Plücker function) \(\mathop{\mathrm{sgn}}\circ \varphi: E^n \longrightarrow \mathbb{S} = \{0,+1,-1\}\). An equivalent definition of an oriented valuated matroid is also given by a compatible pair of a tropical Plücker vector and a chirotope, see [41].

There is a cryptomorphic definition of \(\mathbb{H}\)-matroids via \(\mathbb{H}\)-circuits. For \(\mathbb{H}\in \{\mathbb{K}, \mathbb{S}, \mathbb{T}\}\) these give the usual axioms for circuits, signed circuits and valuated circuits. The general cryptomorphic axiomatization of \(\mathbb{H}\)-circuits can be found in [21]. For \(\mathbb{H}= \mathbb{R}\mathbb{T}\), we will now state the axioms for the set of \(\mathbb{R}\mathbb{T}\)-circuits. For any \(C,C' \in {\mathbb{R}\mathbb{T}}^E\) we define the support of \(C\) as \(\mathop{\mathrm{Supp}}{C}:= \{e \in E \mid C_e \neq 0\}\), and we define the composition \[\begin{align} (C \circ C')_e = \begin{cases} C_e \quad \text{if } |C_e| \geq |C'_e|,\\ C'_e \quad \text{otherwise}. \\ \end{cases} \end{align}\]

We call \(\mathbb{R}\mathbb{T}\)-circuits signed valuated circuits.

In particular, the set of supports of \(\mathcal{C}\) is the set of circuits of the underlying matroid.

2.4 Real Tropical Linear Spaces↩︎

Analogous to the definition of the tropical linear space of a valuated matroid [43], the real tropical linear space \(\mathop{\mathrm{Trop}}_r(\mathcal{M})\) associated to an oriented valuated matroid \(\mathcal{M}\) is defined as follows:

The tropicalization of a linear space over a non-Archimedean field is determined by its associated valuated matroid [43]. If \(K\) is a real closed field with a compatible absolute value \(\lvert \cdot \rvert_K\), then similarly the real tropicalization of a linear space over \(K\) is determined by the associated oriented valuated matroid.

In other words, the real tropicalization of a linear space only depends on the associated oriented valuated matroid.

3 Real Analytification and Tropicalization↩︎

Throughout this section, we fix a real closed, non-Archimedean valued field \(K\) with absolute value \(\lvert \cdot \rvert_K\) that is compatible with the order on \(K\). We recall the real analytification of a variety over \(K\) which was introduced in [10]. We then show that, similarly to the Berkovich analytification, the real analytification of \(\mathbb{P}^n\) can be described as a quotient of \(\mathbb{A}^{n+1,\mathop{\mathrm{an}}}_r\setminus \{0\}\), and hence can be described in terms of signed multiplicative seminorms on a polynomial ring. This gives a real tropicalization map from \(\mathbb{P}_r^{n,\mathop{\mathrm{an}}}\) to \(\mathbb{R}\mathbb{P}^n\) extending the real tropicalization map from affine varieties to closed subvarieties of \(\mathbb{P}^n\).

3.1 The Real Analytification \(\boldsymbol{X_r^{\mathop{\mathrm{an}}}}\)↩︎

In [10] the authors define a real analytification of a \(K\)-variety \(X\). By a \(K\)-variety \(X\) we always mean a reduced, irreducible, separated scheme of finite type. For a point \(p\) in the scheme \(X\), let \(K(p)\) denote its residue field.

Many properties that hold for the Berkovich analytification also hold for the real analytification. For example:

• For any morphism \(f:X\to Y\) of varieties over \(K\), there is a natural induced continuous map \(f_r^{\mathop{\mathrm{an}}}: X_r^{\mathop{\mathrm{an}}}\to Y_r^{\mathop{\mathrm{an}}}\) of the corresponding real analytifications. This assignment is functorial.

• The space \(X^{\mathop{\mathrm{an}}}_r\) is a connected Hausdorff space and if \(X\) is proper over \(K\), it is compact [10].

There is a natural map \(X^{\mathop{\mathrm{an}}}_r\to X^{\mathop{\mathrm{an}}}\) which forgets the last entry. This map is always continuous and proper, but in general neither injective nor surjective [10]. Just like for the classical Berkovich analytification, for an affine \(K\)-variety \(X = \mathop{\mathrm{Spec}}(A)\), the real analytification \(X_r^{\mathop{\mathrm{an}}}\) has a description as the space of signed multiplicative seminorms on the \(K\)-algebra \(A\).

The space of signed multiplicative seminorms is endowed with the coarsest topology that makes the natural evaluation maps \(\mathop{\mathrm{ev}}_f: \left\lvert \cdot \right \rvert^{\text{sgn}} \mapsto |f|^{\mathop{\mathrm{sgn}}}\) for all \(f \in A\) continuous. By [10] for \(X = \mathop{\mathrm{Spec}}(A)\), we have that \(X_r^{\mathop{\mathrm{an}}}\) is the space of signed multiplicative seminorms on \(A\). The map to the Berkovich analytification is given by \(\left\lvert \cdot \right \rvert^{\text{sgn}} \mapsto |\left\lvert \cdot \right \rvert^{\text{sgn}}|\).

In particular, for the affine space \(\mathbb{A}^n = \mathop{\mathrm{Spec}}K[t_1,\ldots,t_n]\), the real Berkovich analytification \((\mathbb{A}^n)^{\mathop{\mathrm{an}}}_r\) is given as the set of signed multiplicative seminorms on \(K[t_1,\dots,t_n]\). Similar to the Berkovich Proj construction, there exists a corresponding construction for the real analytification, hence we can describe \(\mathbb{P}_r^{n,\mathop{\mathrm{an}}}\) in terms of signed multiplicative seminorms. Let \(\left\lvert \cdot \right \rvert^{\text{sgn}}_1\) and \(\left\lvert \cdot \right \rvert^{\text{sgn}}_2\) be points in \(\mathbb{A}^{n+1,\mathop{\mathrm{an}}}_r \setminus \{0\}\), i.e.,non-zero signed multiplicative seminorms on \(K[t_0,\ldots,t_n]\) extending the absolute value on \(K\). We call \(\lvert \cdot \rvert_1^{\mathop{\mathrm{sgn}}}\) and \(\lvert \cdot \rvert_2^{\mathop{\mathrm{sgn}}}\) homothetic, if there exists a constant \(0 \neq c \in \mathbb{R}\) such that for every homogeneous polynomial \(f\) of degree \(d\), we have \(|f|^{\mathop{\mathrm{sgn}}}_1 = c^d |f|^{\mathop{\mathrm{sgn}}}_2\). The following description of \(\mathbb{P}^{n,\mathop{\mathrm{an}}}_r\) is analogous to the classical case.

There is a natural continuous surjective real tropicalization map \[\begin{align} \mathop{\mathrm{trop}}_r: \mathbb{A}^{n+1,\mathop{\mathrm{an}}}_r &\longrightarrow \mathbb{R}^{n+1},\\ \left\lvert \cdot \right \rvert^{\text{sgn}} &\longmapsto (|t_0|^{\mathop{\mathrm{sgn}}},\dots,|t_n|^{\mathop{\mathrm{sgn}}}), \end{align}\] which induces a real tropicalization map \[\begin{align} \mathop{\mathrm{trop}}_{r}: \mathbb{P}_r^{n,\mathop{\mathrm{an}}} &\longrightarrow \mathbb{R}\mathbb{P}^n, \\ \big[ \left\lvert \cdot \right \rvert^{\text{sgn}} \big] &\longmapsto (|t_0|^{\mathop{\mathrm{sgn}}}:\dots:|t_n|^{\mathop{\mathrm{sgn}}}). \end{align}\]

Let \(\iota: X\hookrightarrow \mathbb{P}^n\) be a closed subvariety. By a projective version of the real analogue of the fundamental theorem [10], we have \[\mathop{\mathrm{trop}}_{r}(X_r^{\mathop{\mathrm{an}}}) = \mathop{\mathrm{Trop}}_r(X, \iota)\, .\]

4 The Signed Goldman–Iwahori Space↩︎

The signed Goldman–Iwahori space associated to a vector space can be thought of as a linear analogue of the real analytification of \(\mathbb{P}^n\). Alternatively, it is a version of the Goldman–Iwahori space in [1] that takes signs of the ground field into account. In this section, we will examine its structure and its relation to the real analytification. Further, we describe a real tropicalization map from the signed Goldman–Iwahori space to real projective space. This allows us to define a real tropicalization map from the signed Goldman–Iwahori space to real tropicalizations of linear spaces. From now on, let \(K\) be a real closed field with compatible non-Archimedean absolute value \(\lvert \cdot \rvert\) and let \(V\) be a finite-dimensional vector space over \(K\).

4.1 Signed Seminorms↩︎

In the non-signed case, if the field is spherically complete (or equivalently, maximally complete), then every seminorm on a finite-dimensional vector space is diagonalizable [9]. In particular, this is true for trivially or discretely valued fields, since they are spherically complete. In fact, the reverse implication also holds: If \(\dim V \geq 2\) and \(K\) is not spherically complete, then there exists a seminorm on \(V\) that is not diagonalizable. In the signed world, the situation is more complicated, as the following example will show:

We endow the set of signed seminorms on \(V\) with the topology of pointwise convergence. This is the coarsest topology such that all evaluation maps \(\mathop{\mathrm{ev}}_v: \left\lVert \cdot \right \rVert^{\text{sgn}} \mapsto \left\lVert v \right \rVert^{\text{sgn}}\) for \(v \in V\) are continuous. It agrees with the subspace topology of \(\mathbb{R}\mathbb{P}^{(K^{n+1})^*}\). Two signed seminorms \(\left\lVert \cdot \right \rVert^{\text{sgn}}_1, \left\lVert \cdot \right \rVert^{\text{sgn}}_2\) are said to be homothetic, written \(\left\lVert \cdot \right \rVert^{\text{sgn}}_1 \sim \left\lVert \cdot \right \rVert^{\text{sgn}}_2\), if there is a constant \(c \neq 0\) such that \(\left\lVert \cdot \right \rVert^{\text{sgn}}_1 = c \left\lVert \cdot \right \rVert^{\text{sgn}}_2\). Homothety defines an equivalence relation on the space of signed seminorms.

In [9], the (non-signed) Goldman–Iwahori space is denoted by \(\overline{\mathcal{X}}(V)\), which is a compactification of the space of norms modulo homothety \(\mathcal{X}(V)\). Even though \(\mathfrak{R}\mathcal{X}(V)\) will turn out to be compact (Corollary 1), we omit the bar to declutter notation.

Note the dualization in Definition 7. This makes the assignment \(V\mapsto \mathfrak{R}\mathcal{X}(V)\) a covariant functor from the category of finite-dimensional vector spaces over \(K\) to the category of topological spaces via pulling back signed seminorms under linear maps. For a linear map \(f:V\to W\), we write \(\mathfrak{R}\mathcal{X}(f):\mathfrak{R}\mathcal{X}(V)\to \mathfrak{R}\mathcal{X}(W)\) for the induced continuous map.

4.2 Tropicalization of \(\boldsymbol{\mathfrak{R}\mathcal{X}_n(K)}\)↩︎

From now on, we consider the vector space \(V=K^{n+1}\) together with its defined standard basis \(\mathbf{e} = (e_0,\ldots,e_n)\) and the associated dual basis \(\mathbf{e}^\ast = (e_0^\ast,\ldots, e_n^\ast)\) of \(V^\ast\). This identifies \(\mathbb{A}(V)\) and \(\mathbb{P}(V)\) with \(\mathbb{A}^{n+1}=\mathop{\mathrm{Spec}}K[t_0,\ldots, t_n]\) and \(\mathbb{P}^n=\mathop{\mathrm{Proj}}K[t_0,\ldots, t_n]\), respectively.

We have a natural continuous map \(\tau: \mathbb{P}_r^{n,\mathop{\mathrm{an}}} \xrightarrow{} \mathfrak{R}\mathcal{X}_n(K)\) by restricting a non-zero signed multiplicative seminorm on \(K[t_0,\dots,t_n]\) to its degree \(1\) part \(K[t_0,\dots,t_n]_1 \cong (K^{n+1})^{\ast}\). Note that this is well-defined, taking into account the respective equivalence relations: if \(\left\lvert \cdot \right \rvert^{\text{sgn}}_1 \sim \left\lvert \cdot \right \rvert^{\text{sgn}}_2 \in \mathbb{P}_r^{n,\mathop{\mathrm{an}}}\), then there is \(c\neq 0\) such that for any homogeneous linear polynomial \(f \in K[t_0,\dots, t_n]_1\) we have \(\left\lvert f \right \rvert^{\text{sgn}}_1=c\left\lvert f \right \rvert^{\text{sgn}}_2\), hence the restrictions of \(\left\lvert \cdot \right \rvert^{\text{sgn}}_1\) and \(\left\lvert \cdot \right \rvert^{\text{sgn}}_2\) are homothetic.

Note that the association in Definition 8 only depends on the homothety class of \(\left\lVert \cdot \right \rVert^{\text{sgn}}\), so it indeed descends to a map \(\mathfrak{R}\mathcal{X}_n(K)\to \mathbb{R}\mathbb{P}^n\). Moreover, by the definition of the topology of \(\mathfrak{R}\mathcal{X}_n(K)\), the real tropicalization map \(\mathop{\mathrm{trop}}_{\mathfrak{R}\mathcal{X}_n}\) is continuous. Using Example 8, one can construct an inverse image for each point in \(\mathbb{R}\mathbb{P}^n\) after possibly replacing some basis vectors by their negatives, hence \(\mathop{\mathrm{trop}}_{\mathfrak{R}\mathcal{X}_n}\) is also surjective.

Recall that the real tropicalization map is given by \[\begin{align} \mathop{\mathrm{trop}}_{r}: \mathbb{P}^{n,\mathop{\mathrm{an}}}_r &\longrightarrow \mathbb{R}\mathbb{P}^n\, ,\\ \big[\left\lvert \cdot \right \rvert^{\text{sgn}}\big] &\longmapsto \big(|t_0|^{\mathop{\mathrm{sgn}}}:\dots:|t_n|^{\mathop{\mathrm{sgn}}}\big)\, . \end{align}\] By construction, \(\mathop{\mathrm{trop}}_r:\mathbb{P}_r^{n,\mathop{\mathrm{an}}}\to \mathbb{R}\mathbb{P}^n\) factors as \[\label{eq95factorization} \mathbb{P}_r^{n,\mathop{\mathrm{an}}}\xlongrightarrow{\tau}\mathfrak{R}\mathcal{X}_n(K)\xlongrightarrow{\mathop{\mathrm{trop}}_{\mathfrak{R}\mathcal{X}_n}} \mathbb{R}\mathbb{P}^n \, .\tag{1}\] Let \(\iota = (f_0:\dots:f_m): \mathbb{P}^n\hookrightarrow \mathbb{P}^m\) be a linear embedding, where \(f_i \in (K^{n+1})^*\). We define \[\pi_{\iota}\mathrel{\vcenter{:}}= \;\mathop{\mathrm{trop}}_{\mathfrak{R}\mathcal{X}_m}\circ\;\mathfrak{R}\mathcal{X}(\iota): \mathfrak{R}\mathcal{X}_n(K)\longrightarrow \mathbb{R}\mathbb{P}^m\, .\] A direct computation shows that for all \(\big[\left\lVert \cdot \right \rVert^{\text{sgn}}\big] \in \mathfrak{R}\mathcal{X}_n(K)\), we have \[\pi_{\iota}\big(\big[\left\lVert \cdot \right \rVert^{\text{sgn}}\big]\big) = \big(||f_0||^{\mathop{\mathrm{sgn}}}:\ldots : ||f_m||^{\mathop{\mathrm{sgn}}}\big)\, .\]

5 The Limit Theorem↩︎

In this section, we show a linear version of the limit theorem [10] which is a real analogue of the limit theorem [9]. We will have a similar setup as in [9] which we now recall. We first set up a category of linear embeddings such that real tropicalization yields a covariant functor into the category of topological spaces.

::: center
:::

In particular, there is a natural map
$$\varprojlim_{\iota \in I} \pi_{\iota}: \mathfrak{R}\mathcal{X}_n(K) \to \varprojlim_{\iota \in I} \mathop{\mathrm{Trop}}_r\big(\mathbb{P}^n,\iota\big).$$

5.1 Relation to the Goldman–Iwahori Space↩︎

Recall that the Goldman–Iwahori space \(\overline{\mathcal{X}}_n(K)\) is the space of non-trivial (ordinary) seminorms on \((K^{n+1})^*\) modulo homothety. It has been studied first by Goldman and Iwahori [1] (also see for more recent accounts [5], [7], [15]). However, our notation will follow [9], where it was proved that \(\overline{\mathcal{X}}_n(K)\) is the limit of all linear tropicalizations of \(\mathbb{P}^n\). The compactified affine Bruhat–Tits building \(\overline{\mathcal{B}}_n(K)\) of \(\mathop{\mathrm{PGL}}((K^{n+1})^*)\) is the subspace of \(\overline{\mathcal{X}}_n(K)\) of classes of diagonalizable seminorms. The inclusion \(\overline{\mathcal{B}}_n(K)\subseteq \overline{\mathcal{X}}_n(K)\) is dense by [9] and we have an equality if and only if \(K\) is spherically complete by [46].

The association \(\left\lVert \cdot \right \rVert^{\text{sgn}} \mapsto \big|\left\lVert \cdot \right \rVert^{\text{sgn}}\big|\) defines a natural map \[\Phi: \mathfrak{R}\mathcal{X}_n(K) \longrightarrow \overline{\mathcal{X}}_n(K)\, .\] Since morphisms in \(I\) commute with taking pointwise absolute values, we obtain the following:

In general, the fibers of \(\Phi\), though non-empty, can vastly differ in complexity. In Section 7, we will consider the trivially valued case \(K=\mathbb{R}\) and show that the fibers can be singletons, finite sets, or even infinite.

6 The Universal Realizable Oriented Valuated Matroid↩︎

We may extend the definition of a matroid over a hyperfield (Definition 1) to infinite ground sets. This generalization to infinite ground sets works as in [9] which is done for the tropical hyperfield \(\mathbb{T}\), i.e., for valuated matroids of finite rank on possibly infinite ground sets. In this section, we will do the same for the real tropical hyperfield \(\mathbb{R}\mathbb{T}\).

Let \(E\) be now a possibly infinite ground set and \(\mathcal{M}\) be an oriented valuated matroid given by a Grassmann–Plücker function \(\varphi: E^n \to \mathbb{R}\mathbb{T}\). Note that the rank of \(\mathcal{M}\) is \(n\), so it is in particular finite. We define \(\mathbb{R}\mathbb{P}^E : = \{(y_e)_{e\in E} | y_e \in \mathbb{R}\} \setminus \{(0)_{e \in E}\} / \mathbb{R}^*\) and equip it with the topology of pointwise convergence. Analogous to Definition 3 we define:

Recall that \(\oplus\) denotes the hyperfield sum of the real tropical hyperfield \(\mathbb{R}\mathbb{T}=\mathbb{R}\).

Let \(K\) be a real closed field with a compatible absolute value \(\lvert \cdot \rvert_K\). We now extend the construction of realizable oriented valuated matroids from Example 7 to the ground set \(E = (K^{n+1})^*\).

By considering \(- \log |\varphi_{\mathop{\mathrm{univ}}}|\), we obtain the universal realizable valuated matroid \(w_{\mathop{\mathrm{univ}}} = \mathop{\mathrm{val}}\circ \det\) studied in [9]. Hence, in contrast to \(w_{\mathop{\mathrm{univ}}}\), the map \(\varphi_{\mathop{\mathrm{univ}}}\) also keeps track of the signs.

Note that both the signed Goldman–Iwahori space and the real tropical linear space associated to the universal realizable oriented valuated matroid are defined to be subsets of \(\mathbb{R}\mathbb{P}^E\).

7 The Case of Real Numbers with Trivial Valuation↩︎

For the case that \(K=\mathbb{R}\) with trivial valuation, it will turn out that all signed seminorms are diagonalizable. This lets us describe the signed Goldman–Iwahori space explicitly over this field. Moreover, it allows us to describe the real tropical linear space of \(\mathcal{M}_{\mathop{\mathrm{univ}}}\) as a real Bergman fan.

7.1 Diagonalizability of Signed Seminorms↩︎

We want to describe \(\mathfrak{R}\mathcal{X}(V)\) explicitly by making use of the map \(\Phi: \mathfrak{R}\mathcal{X}(V)\to \overline{\mathcal{B}}(V)\), which assigns to each signed seminorm its absolute value. Therefore, we quickly recall an explicit description of \(\overline{\mathcal{B}}(V)\) in terms of flags of subspaces. The following holds for any trivially valued field \(K\):

The bijection works the following way: for a homothety class of seminorms we choose the representative \(\lVert \cdot \rVert\) that has maximal value \(1\) on \(V\). Then one obtains the flag of subspaces as subsets \(\lVert \cdot \rVert^{-1}(0,\varepsilon)\) by letting \(\varepsilon\) vary. The kernel of \(\lVert \cdot \rVert\) then equals \(V_0\) and the coordinates \(d_i\) are given by the constant value of \(\lVert \cdot \rVert\) on \(V_i\setminus V_{i-1}\) for \(i=1,\dots,l-1\). In particular, \(\overline{\mathcal{B}}(V)\) can be identified with the compactified cone over the order complex of the lattice of non-trivial subspaces of \(V^{\ast}\).

From now on, we again consider \(K=\mathbb{R}\). Preimages of intervals in the signed case are no longer subspaces, but only convex cones:

where $A^{\circ}/\overline{A}/\partial A$ denote the
interior/closure/boundary of $A$ in $V$ with respect to the
Euclidean topology.

We note that \(K = \mathbb{R}\) is the only real closed field that admits the separation Lemma [48]. Thus for an arbitrary real closed field \(K\) we cannot expect a hyperplane \(\langle \cdot,w \rangle = 0\) separating \(A\) and \(-A\) to exist.

This now allows us to proof, that every signed seminorm is diagonalizable:

Diagonalizability gives us a description of \(\mathfrak{R}\mathcal{X}(V)\) in the flavor of Proposition 23.

We consider two signed flags equivalent, if the underlying flag of subspaces is the same and the choice of region is exactly opposite for all \(i\). The notion of signed flags helps us to formulate a signed analogue of Proposition 23.

Together, Proposition 23 and Proposition 26 allow us to study the natural map \(\Phi: \mathfrak{R}\mathcal{X}(V)\to \overline{\mathcal{B}}(V)\). We want to understand \(\Phi([\left\lVert \cdot \right \rVert^{\text{sgn}}])\) for some \([\left\lVert \cdot \right \rVert^{\text{sgn}}]\in \mathfrak{R}\mathcal{X}(V)\). Let \(0\subseteq V_0 \subsetneq V_1 \subsetneq \dots \subsetneq V_l=V^*\) be the corresponding signed flag and \(0\leq d_1\leq \dots\leq d_{l-1} \leq 1\) constructed as in Proposition 26. Let \(i_0\mathrel{\vcenter{:}}= 0\) and \(0<i_1\leq \dots \leq i_{l'} <l\) be the indices such that there is a strict inequality \(d_{i_j}<d_{i_j+1}\) (here we set and \(d_l \mathrel{\vcenter{:}}= 1\)). Set \(V'_j\mathrel{\vcenter{:}}= V_{i_j}\) for \(j=1,\dots,l\) and \(d'_j=d_{i_j}\). Then \(\Phi\left(\big[\left\lVert \cdot \right \rVert^{\text{sgn}}\big]\right)\) is given by the flag \(0=V'_0 \subsetneq V'_1 \subsetneq \dots \subsetneq V'_{l'} \subsetneq V'_{l'+1} \mathrel{\vcenter{:}}= V^*\) and coordinates \(0\leq d'_1<\dots< d'_l<1\) as in Proposition 23.

A natural question to ask is: what do the fibers of \(\Phi\) look like? This has an easy answer, if the flag that is induced by the homothety class of a seminorm is complete, i.e.jumps of subspaces are only by one dimension. In that case, all preimages are given by the same coordinates \(d_i\) and any choice of signature on the flag yields a signed seminorm, hence the fiber under \(\Phi\) has exactly \(2^{n-1}\) elements. The situation gets more complicated, if the flag is not complete, as the following example will show.

Example 11. Consider the case \(V=\mathbb{R}^2\). As explained in Example 3, there are three types of points of \(\overline{\mathcal{B}}(V)\):

(a) the homothety class \(\eta\) of the constant norm,

(b) classes of proper norms with \(2\) different non-zero values,

(c) classes of proper seminorms (i.e., seminorms with non-trivial kernel).

If \(K\) was algebraically closed, we would have \(\overline{\mathcal{B}}_1(K)=\mathbb{P}^{1,\mathop{\mathrm{an}}}\) and these would exactly correspond to Berkovich type II, type III, respectively type I points of the space. However, \(\mathbb{R}\) is not algebraically closed and thus \(\overline{\mathcal{B}}_1(\mathbb{R}) \subsetneq \mathbb{P}^{1,\mathop{\mathrm{an}}}\).

As discussed before, if a point in \(\overline{\mathcal{B}}_1(\mathbb{R})\) is of type (b), then its fiber consists of two points. If the point is of type (c), then there is only one preimage, as the two choices of signed flags are equivalent. In contrast, the fiber of \(\eta\) is quite large. For every line \(V_1 \subset \mathbb{R}^2\), there are two equivalence classes of signed flags with underlying flag \(0 \subsetneq V_1 \subsetneq \mathbb{R}^2\). Hence, there is a (non-canonical) bijection \[\Phi^{-1}(\eta) \cong \mathbb{P}^1(\mathbb{R}) \sqcup \mathbb{P}^1(\mathbb{R})\, .\] Note that this bijection is not continuous, so it does not tell us anything about the topology. To find out more about the topology of this fiber we refer to [10], since in the special case of dimension \(2\), one can show that \(\overline{\mathcal{B}}_1(\mathbb{R})\) is the set of real points of \(\mathbb{P}^{1,\mathop{\mathrm{an}}}\) and \(\mathfrak{R}\mathcal{X}_1(\mathbb{R})\) equals \(\mathbb{P}^{1,\mathop{\mathrm{an}}}_r\). In Figure 5 we sketch the map \(\Phi\). Note that because of the infinite nature of the spaces, the topology on both \(\mathfrak{R}\mathcal{X}_1(\mathbb{R})\) and \(\overline{\mathcal{B}}_1(\mathbb{R})\) is coarser than the figure might lead us to believe. For example, the space \(\mathfrak{R}\mathcal{X}_1(\mathbb{R})\) is connected (this follows from 27), even though as a set it is a disjoint union of closed intervals.

8 Real Bergman Fans↩︎

The tropical linear space associated to a matroid has the structure of a fan, called the Bergman fan. Recall that to each oriented valuated matroid one can associate a real tropical linear space. If the valuation is trivial, i.e., we have an oriented matroid, the corresponding fan is called the real Bergman fan. This construction in the case of finite ground sets was introduced in [25]. We will first discuss circuit axiomatization of infinite oriented matroids and then recall the notion of covectors of an oriented matroid, which are then used to define its real Bergman fan. In this section, we consider \(K\) to be a real closed and trivially valued field.

8.1 Infinite Oriented Matroids↩︎

A signed subset is a function \(C:E\to \{0,+1,-1\}\). We denote by \(C^+\mathrel{\vcenter{:}}= \{e\in E \mid C(e)=+1\},\;C^-\mathrel{\vcenter{:}}= \{e\in E\mid C(e)=-1\}\), \(\mathop{\mathrm{Supp}}(C)\mathrel{\vcenter{:}}= C^+\cup C^-\), and \(C^0 \mathrel{\vcenter{:}}= E \setminus \mathop{\mathrm{Supp}}(C)\). The separation set of two signed sets \(C,C'\) is defined by \(S(C,C')\mathrel{\vcenter{:}}= \{e\in E \mid C(e)=-C'(e)\neq 0\}\).

We quickly discuss the circuit axiomatization of infinite oriented matroids of finite rank. Fix a possibly infinite ground set \(E\). Recall that an oriented matroid is given by a chirotope (i.e., a Grassmann–Plücker function) \(\varphi: E^n\to \mathbb{S}=\{0,+1,-1\}\). Equivalently, there is the same circuit axiomatization as in Proposition 6 (replacing \(\mathbb{R}\mathbb{T}\) with \(\mathbb{S}\)), with one additional axiom, which ensures finite rank:

• There is a positive integer \(n \in \mathbb{N}\) such that for all subsets \(A\subseteq E\) with \(|A|>n\) there exists \(C\in \mathcal{C}\) with \(\mathop{\mathrm{Supp}}(C)\subseteq A\).

By restricting and applying the finite case (e.g., [19]) one obtains the strong circuit elimination:

• For all \(C\neq -C'\in \mathcal{C}\), \(e \in S(C,C')\), and \(f\in (C^+ \setminus C'^-)\cup (C^-\setminus C'^+)\), there is \(C'' \in \mathcal{C}\) such that \(C''(e)=0\), \(f\in \mathop{\mathrm{Supp}}(C'')\), and \[\begin{align} C''^+ \subseteq(C^+\cup C'^+), \quad C''^- \subseteq(C^-\cup C'^-). \end{align}\]

As a consequence, our notion of oriented matroids agrees with the characterization given in [49] with the additional requirement of finite rank.

8.2 Covectors↩︎

Let \(\mathcal{M}\) be an oriented matroid given by a chirotope \(\varphi:E^n\to \{0,+1,-1\}\).

The composition of two signed sets \(C,C':E\to \{0,+1,-1\}\) is defined by \[(C\circ C')(e)=\begin{cases} C(e) \quad \text{ if } C(e)\neq 0,\\ C'(e) \quad \text{if } C(e)= 0.\\ \end{cases}\] We write \(C\leq C'\) if \(C'^0 \subseteq C^0\) and \(C'|_{\mathop{\mathrm{Supp}}C}=C|_{\mathop{\mathrm{Supp}}C}\). In other words, this partial order is induced by the partial order \(0 < +1,-1\), considered coordinatewise.

If \(E\) is finite, the covectors of the oriented matroid are exactly the vectors of its dual [19].

Vice versa, let $\left\lVert \cdot \right \rVert^{\text{sgn}}$ be a
signed seminorm. Then by Proposition
[24](#prop:norm95diagonalizable){reference-type="ref"
reference="prop:norm95diagonalizable"} it is diagonalizable by an
ordered basis $B=(b_0,\dots,b_n)$ with vector
$\vec{c}=(c_0,\dots,c_n)$. Let $B_i=((-1)^ib_i,b_1,\dots,b_n$ and
$\vec{c}_i\mathrel{\vcenter{:}}= (c_i,0,\dots,0,\dots,0)$. Then
$\left\lVert \cdot \right \rVert^{\text{sgn}}_{B,\vec{c}}=\left\lVert \cdot \right \rVert^{\text{sgn}}_{B,\vec{c}_0} \circ \dots \circ \left\lVert \cdot \right \rVert^{\text{sgn}}_{B,\vec{c}_n}$.
As before we have that
$\mathop{\mathrm{sgn}}(\left\lVert \cdot \right \rVert^{\text{sgn}}_{B,\vec{c}_i})$
is a cocircuit, hence the composition is a covector.

If \(E\) is infinite, the covector axiom (Cov4) need not be true. For example, one can show that the covector poset of the restriction of \(\mathcal{M}_{\mathop{\mathrm{univ}}}\) to \(E'\) in Example 14 does not fulfill (iv).

The restriction \(\mathcal{M}|_{E'}\) of an oriented matroid to a subset \(E'\subseteq E\) is the oriented matroid whose signed circuits are the signed subsets of \(\mathcal{C}\) whose support is contained in \(E'\). One can easily check that the axioms (C0)-(C4) hold and that restriction commutes with taking the underlying matroid. This is equivalent to restricting a chirotope.

Consequences of the absence of duality leads to different behavior of the covector poset:

8.3 Real Bergman Fans↩︎

Let \(\mathcal{M}\) be an oriented matroid of rank \(n+1\) on any ground set \(E\) with poset of covectors \(\mathop{\mathrm{Cov}}\). We define its real Bergman fan \(\Sigma^*_{\mathcal{M}}\) to be given by the collection of cones \[\langle X_1, \dots X_l\rangle_{\mathbb{R}_{> 0}} \subset \mathbb{R}\mathbb{P}^E\] for each chain of non-zero covectors \(X_1< \dots < X_l\). This is, as in the non-oriented case, a fan of pure dimension \(n\). If \(E\) is infinite, \(\Sigma^*_{\mathcal{M}}\) may have infinitely many cones and is embedded into an infinite-dimensional real projective space.

For finite oriented matroids, the notions of its real Bergman fan and its associated real tropical linear space agree:

::: center
:::

If now $\iota: \mathbb{P}^n\to \mathbb{P}(\mathbb{R}^{E'})$ is the
corresponding embedding, we obtain commutativity for

::: center
:::

Outlook and applications↩︎

Several lines of further inquiry naturally extend the framework of this paper.

Reductive groups over real closed fields. The signed Goldman–Iwahori space provides a real-analytic, non-Archimedean analogue of a symmetric space for \(\mathop{\mathrm{PGL}}\) over real closed fields. It should be a natural starting point for a signed analogue of Bruhat–Tits theory for reductive groups over real closed valued fields, paralleling the role of \(\overline{\mathcal{X}}_n(K)\) in the non-Archimedean theory [5]–[7].

The non-realizable case. The limit theorem (Theorem 1) ranges over linear embeddings of \(\mathbb{P}^n\), equivalently, over realizable oriented valuated matroids, and Theorem 2 identifies the limit with the real tropical linear space of the universal realizable oriented valuated matroid. The construction of \(\mathop{\mathrm{Trop}}_r(\mathcal{M})\) in Section 6, however, applies to arbitrary finite-rank oriented valuated matroids, and it is natural to ask what space arises as the inverse limit of \(\mathop{\mathrm{Trop}}_r(\mathcal{M})\) over a suitable category of all (possibly non-realizable) such matroids. A description of this space would be interesting to study.

Real toric vector bundles. A classical result of Klyachko, recast tropically by Kaveh and Manon [52], classifies \(T\)-equivariant vector bundles on a toric variety \(X_\Sigma\) in terms of piecewise \(\mathbb{Z}\)-linear maps from the fan \(\Sigma\) to the building \(B(\mathop{\mathrm{PGL}}_n(K))\). A non-trivially valued generalization can be found in [53]. Over a real closed field \(K\), the signed Goldman–Iwahori space \(\mathfrak{R}\mathcal{X}_n(K)\) is the natural replacement for the building, and it would be interesting to ask whether real toric vector bundles on \(X_\Sigma\) admit an analogous classification by piecewise linear maps \(\Sigma \to \mathfrak{R}\mathcal{X}_n(K)\), with the signed data encoding the real structure beyond the underlying complex bundle.

References↩︎