Buildings, valuated matroids, and tropical linear spaces

Affine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL$\mathrm{PGL}$ parameterizes flags of subspaces/lattices in or, equivalently, norms on a fixed finite‐dimensional vector space, up to homothety. It has first been studied by Goldman and Iwahori as a piecewise‐linear analogue of symmetric spaces. The space of seminorms compactifies the space of norms and admits a natural surjective restriction map from the Berkovich analytification of projective space that factors the natural tropicalization map. Inspired by Payne's result that the analytification is the limit of all tropicalizations, we show that the space of seminorms is the limit of all tropicalized linear embeddings ι:Pr↪Pn$\iota \colon \mathbb {P}^r\hookrightarrow \mathbb {P}^n$ and prove a faithful tropicalization result for compactified linear spaces. The space of seminorms is in fact the tropical linear space associated to the universal realizable valuated matroid.


Introduction
Let K be a complete non-Archimedean field (possibly carrying the trivial absolute value).The non-Archimedean analytic approach to tropical geometry (see e.g.[BPR16,EKL06,Gub13,Pay09]) makes crucial use of non-Archimedean analytic spaces in the sense of Berkovich [Ber90]; the particular topological properties of these Berkovich spaces let us think of the analytification X an of an algebraic variety X as a form of universal tropicalization that is independent of the chosen coordinate system.In this regard, the central result presented in [Pay09] tells us that the analytification X an of a quasi-projective algebraic variety X over K is the projective limit of all tropicalizations with respect to all the embeddings in toric varieties (also see [FGP14,GG22,KSU21] for generalizations of this result).
In this article we argue that when considering only the tropicalization of projective spaces linearly embedded into higher-dimensional projective spaces, the role of Berkovich analytic space is played by the Goldman-Iwahori space X r (K) of homothety classes of non-trivial seminorms on (K r+1 ) * (see [GI63] but also [RTW15,Wer04,RTW12]).The locus B r (K) of diagonalizable seminorms in X r (K) is (a compactification of) the affine Bruhat-Tits building of PGL r+1 (K), when K carries a non-trivial valuation, and the cone over the spherical building of PGL r+1 (K), when the valuation on K is trivial.When K is spherically complete, we have B r (K) = X r (K).
Let ι : P r ֒→ P n be a linear closed immersion.The tropicalization Trop(P r , ι) of P r with respect to the embedding ι is the projection of ι(P r ) an ⊆ (P n ) an under the natural tropicalization map trop P n : (P n ) an −→ TP n to TP n = R n+1 − (∞, . . ., ∞) / R • ½ that is essentially given by taking coordinate-wise valuations (see Section 2 below for details).The tropicalization Trop(P r , ι) is a projective tropical linear space in TP n and is associated to a realizable valuated matroid (see Section 4 below for details).
We shall see in Section 2 below that there is a natural continuous and surjective restriction map τ : (P n ) an → X n (K) that factors the tropicalization map as such that the tropicalization of Trop(P r , ι) is given as the projection of X r (K) ⊆ X n (K) under trop X n .The connection between affine buildings and Berkovich analytic spaces is well-established in the literature.We refer the reader in particular to [Ber90] as well as to [RTW10,RTW12,RTW15].
Denote by I the category whose objects are linear closed immersions ι : P r ֒→ U ⊆ P n into a torusinvariant open subset U ⊆ P n ; a morphism between ι : P r ֒→ U ⊆ P n and ι ′ : P r ֒→ U ′ ⊆ P n ′ is given by a linear toric morphism ϕ : U → U ′ such that ϕ • ι = ι ′ .The tropicalization with respect to ι : P r ֒→ U is naturally homeomorphic to the tropicalization with respect to the composition P r ֒→ U ⊆ P n .For a toric morphism ϕ : U → U ′ such that ϕ • ι = ι ′ , we have a natural induced map ϕ trop : U trop → U ′ trop such that ϕ trop Trop(P r , ι) ⊆ Trop(P r , ι ′ ), making I into a cofiltered category.Requiring morphisms only to be defined on an open subset of projective space provides us with the added flexibility that we need in the following.
Theorem A. The tropicalization maps induce a natural homeomorphism where the projective limit is taken over the category I.
Let I ′ be the full subcategory of I, whose objects are linear embeddings ι : P r ֒→ U ⊆ P n whose images meet the big torus G n m ⊆ P n .Then Theorem A provides us with a homeomorphism between the space of norms X r (K) and the projective limit of all non-compactified tropicalizations Trop ι(P r ) ∩ G n m .When K is spherically complete, the Goldman-Iwahori space X r (K) is equal to B r (K), and therefore we have a natural homeomorphism Trop ι(P r ) ∩ G n m .
In addition to Theorem A we also prove the following, which one may think of as a new addition to the literature on faithful tropicalization (see e.g.[BPR16, CHW14, GRW16, GRW17] for other instances).
Theorem B. Let ι : P r ֒→ P n be a linear closed immersion.Then there is a natural piecewise linear embedding J : Trop(P r , ι) → B r (K) that makes the following diagram commute B r (K) Trop(P r , ι) Suppose now that K is discretely valued.Then, for one, the section in Theorem B recovers the socalled membrane of a realization of Trop(P r , ι) from [JSY07, Lemma 4].If K is also local, there is a natural embedding of B r (K) = X r (K) into the Berkovich analytic space (P r ) an (see [RTW12, Section 3] for details).Theorem A then tells us that the collection of all linear reembeddings P r ֒→ P n recovers exactly B r (K) = X r (K).The main result of [Pay09], on the other hand, tells us that, once we also allow non-linear algebraic re-embeddings of P r into suitable toric varieties, we recover the whole Berkovich analytification of P r .
Theorem A and Theorem B together provide us with a heuristic saying that X r (K) is in some sense a universal (realizable) tropical linear space.This goes hand in hand with the work of Dress and Terhalle [DT98], in which the authors realize the building B r (K) as the tight span of a suitable infinite valuated matroid, see Section 7.2 below.In Section 7.1 we build on this observation and construct a tropical linear space for any infinite valuated matroid (expanding on the finite case, see [Spe08,SS04] as well as [MS15,Chapter 4]).This allows us to make precise the idea that the Goldman-Iwahori space is the tropical linear space of the universal realizable valuated matroid w univ that is given by the map induced by the permutation-invariant map val • det : Theorem C. Let w univ be the universal realizable valuated matroid.Then X r (K) = Trop(w univ ).
Our approach, in particular, provides us with a notion of tropicalization for a finite-dimensional linear space embedded into an infinite-dimensional space, see again Section 7.1 for details.Theorem C is also, in spirit, very similar to the universal (possibly non-linear) tropicalization of [GG22], which may be identified with Berkovich analytic space.
It is well-known that Bruhat-Tits buildings also admits a simplicial structure with can be described in terms of filtrations by linear subspaces, when K carries the trivial absolute value, and by sublattices, when K carries a non-trivial discrete absolute value.We incorporate those perspectives into our story in Examples 1.10 and 1.12, while, in Section 4.5, we recall the analogous story how valuated matroids over a field with trivial absolute value are nothing but matroids.In Section 6 we illustrate our main results using this alternative point of view on buildings.
Conventions.We write R for R ∪ {∞} with tropical operations min as well as +, and , where ∼ is the equivalence relation generated by translation by a real multiple of the vector ½ = (1, . . ., 1).We write [n] for the set {0, . . ., n} and E r for the set of subsets of cardinality r of a given set E. By K we denote a complete non-Archimedean field, with ring of integers O K and residue field k; in the discretely valued case, O K is Noetherian and its maximal ideal is generated by one element, called a uniformizer and denoted by π.
Acknowledgements.This collaboration was initiated while we were preparing for and participating in a learning seminar, the GAUS-AG on buildings; we thank all other participants, in particular Jiaming Chen, Andreas Gross, Johannes Horn, Lucas Gerth, Jérôme Poineau, Felix Röhrle, Pedro Souza, and Jakob Stix.Particular thanks are due to Annette Werner for answering our countless questions about buildings on several occasions.We also thank Jeffrey Giansiracusa, Marvin Hahn, Diane Maclagan, Anne Parreau, Bertrand Rémy, Victoria Schleis, Petra Schwer, and Jonathan Wise for helpful conversations at various (pre-)stages of this project.

The affine building and its compactification
Let K be a complete non-Archimedean field.In this section, we recall some fundamentals about the space of seminorms on a finite-dimensional vector space V over K.The (compactification of the) affine building of PGL n is the a priori proper subset of diagonalizable (semi)norms.If K is spherically complete these two spaces will turn out to be the same.
Let V be a vector space over K of dimension n.A valued field extension L/K is said to be immediate if it has the same value group and the same residue field.A field is called maximally complete if it does not admit any proper immediate extension.By a classical result of Kaplansky (see e.g.[Sch50], II.6, Theorem 8), it turns out that this concept is equivalent to spherical completeness.We thus have the following: Theorem 1.4.[Kru32, Satz 24] Every non-Archimedean field admits a (non-unique) immediate extension that is spherically complete.
The following observation will be central for our result: Proposition 1.5.Let K be a non-Archimedean field.Then K is spherically complete if and only if every seminorm on a finite-dimensional K-vector space is diagonalizable.
Proof.By [BE21, Lemma 1.12] the non-Archimedean field K is spherically complete if and only if all norms on a finite-dimensional vector space over K are diagonalizable.Our statement follows from the fact that a seminorm is diagonalizable if and only if the induced norm on the quotient modulo the kernel is diagonalizable.
(ii) Every discretely valued complete field is spherically complete, as by [RTW12, Proposition 3.1] every norm on a finite-dimensional vector space is diagonalizable.(iii) In particular, for any prime number p the field of p-adic numbers Q p is local and spherically complete.(iv) The algebraic closure Q a p of the p-adic numbers is not complete (see [Rob00, Section 3.1.4]).(v) The completion C p of Q a p is still algebraically closed, but not spherically complete.(vi) Over C, the field of rational functions C(t) is not Cauchy complete, as the Cauchy sequence n i=0 1 i! t i n∈N has no limit.Its completion is the field of (formal) Laurent series C((t)), which is spherically complete, because the valuation is discrete.(vii) For any field k let k{{t}} = n≥1 k((t 1/n )) be the field of Puiseux series.It is not complete.Its completion is the Levi-Civita field [BCS18, Theorem 4.10], in which the exponents of a series need not have a common denominator, but for any upper bound, there are only finitely many exponents with non-zero coefficients.The Levi-Civita field is not spherically complete, its spherical completion is the Malcev-Neumann field k((t Q )) of power series with rational exponents and well-ordered support (see [BE21, Example 1.1]).
Let N (V ) and N (V ) be the set of norms and respectively non-trivial seminorms on the dual vector space V * (note the dualization!).For x ∈ N (V ) we denote by ||.|| x the associated seminorm.We endow N (V ) with the coarsest topology that makes the natural evaluation maps for all v ∈ V * continuous.The space N (V ) has been first introduced by Goldman and Iwahori in [GI63] over K = Q p .
Remark 1.7 (Topology of N (V )).(i) Equivalently, one may define the topology on N (V ) as the topology of pointwise convergence: a net (x α ) in N (V ) converges to a seminorm x ∈ N (V ) if and only if v xα converges to v x in R for all v ∈ V * .(ii) A basis of the topology on N (V ) is given by open subsets of the form for some v 1 , . . ., v l ∈ V * and open intervals (a 1 , b 1 ), . . ., (a l , b l ) ⊆ R, where ev denotes the natural evaluation map.
Two points x, y ∈ N (V ) are said to be homothetic, written as x ∼ y, if there is a constant c > 0 such ||.||x = c • ||.|| y .Homothety defines an equivalence relation on N (V ) that restricts to an equivalence relation on N (V ).
We denote by N diag (V ) ⊆ N (V ) the subspace of diagonalizable norms and N diag (V ) ⊂ N (V ) the subspace of diagonalizable seminorms.Note that seminorms that are homothetic to a diagonalizable one are themselves diagonalizable.Let X (V ) = N (V )/ ∼ .
Definition 1.8.The affine Bruhat-Tits building of PGL(V ) is defined to be the quotient space of diagonalizable norms by homothety: by Corollary 2.8 below X (V ) is compact).In particular, for a spherically complete field K the quotient B(V ) = X (V ) is a natural compactification of B(V ).This expands on the construction in [Wer04].When V = K n+1 , we write X n (K) and X n (K) for X (K n+1 ) and X (K n+1 ) respectively, similarly B n (K) and B n (K).
Remark 1.9.(Topological structure) (i) By [BE21, Theorem 1.19] the locus N diag (V ) of diagonalizable seminorms is dense in N (V ) with respect to the Goldman-Iwahori metric The finiteness of the supremum follows from the fact that any two norms are equivalent (see [BE21, Page 10]).The topology induced by this metric is finer than the one defined above since uniform convergence implies pointwise convergence.It follows that B(V ) is dense in X (V ).Hence also B(V ) is dense in X (V ).Indeed, a stratum of B(V ) of seminorms with a fixed kernel W ⊂ V can be identified with B(V /W ).(ii) Despite the notation suggesting that N (V ) ⊂ N (V ) is open, this need not be true in general, as we will show in Remark 2.10.On the other hand, it is when K is local.Indeed, let {x α } α be a net of seminorms, converging to x, and let {v α } α be a net of unit vectors (with respect to some fixed norm on V ) such that ||v α || xα = 0. Since the field is local, the unit sphere is compact and so we can find a convergent subnet converging to v = 0. We conclude that ||v|| x = 0 by continuity and thus x is also a proper seminorm.
Example 1.10.Let K have trivial valuation.Then we can give an explicit description of the space of seminorms up to homothety on a vector space V of dimension r + 1.First recall that by Example 1.6 Item (i) the field K is spherically complete and hence by Proposition 1.5 we have X (V ) = B(V ).We claim that there is a bijection: , We allow the special case of l = 1 where we just have the flag (0 V * ) without any coordinates.
Proof.Let • ∈ N (V ) be a representative of x ∈ B(V ), i.e. a non-trivial seminorm on V * .Then all balls around 0 in V * are subspaces.By letting the radius vary, we obtain a unique filtration and the values are strictly increasing.Let d i denote the constant value on V i \ V i−1 and set for i = 1, . . ., l − 1 Note that any seminorm homothetic to ||.|| yields the same filtration of subspaces and only multiplies all the d i simultaneously by a common scalar.This does not change the c i and thus we obtain a well-defined map.
Vice versa, every flag of subspaces This can be identified with the cone over the order complex of the lattice of nontrivial subspaces of V * .The latter comes with a natural weak topology, where a set is open if and only if its intersection with each cone is relatively open in that cone.This turns this set into the colimit of all cones corresponding to a fixed filtration.However, the topology on the building is much coarser than the weak topology of the cone complex, as the following example will show.
Example 1.12.We consider the case where K is any infinite field with trivial valuation.Then we can identify B 1 (K) with the set: Here η := (0 (K 2 ) * ) corresponds to the homothety class of a norm that is constant away from 0. Each cone corresponds to a one-dimensional subspace and the point at infinity of each cone corresponds to the homothety class of a proper seminorm.In rank one, the homothety class of a proper seminorm is uniquely determined by its kernel, and thus we can identify B 1 (K) \ B 1 (K) with P 1 (K).
A basis of the topology of B 1 (K) is given as follows: choose finitely many vectors v 1 , . . ., v l ∈ (K 2 ) * and intervals (a 1 , b 1 ), . . ., (a l , b l ) ⊂ R and define In particular, if such a set is a neighbourhood of η, then it contains all cones corresponding to subspaces which do not contain any vector v 1 , . . ., v l , i.e. all but finitely many.This is of course not necessarily true for open neighborhoods of η in the weak topology of the cone complex.
The coordinate c is a positive real number.A norm in the homothety class corresponding to (F , c) has generic value 1, and value e −c on V 1 \ {0}.In the case of c = ∞, we have a proper seminorm with kernel V 1 .
The space B(V ) has the structure of an affine building in the sense of [RTW15, Definition 1.9] (see e.g.[Par00, III.1.2]for a proof).We refer the reader also to [BS14] for a discussion of the various axiom systems for affine buildings over possibly non-discretely valued fields.For us, however, the most important notion is the following: Every apartment is a closed subset homeomorphic to R n+1 /R ≃ R n .The above parametrization can be extended to R n+1 − (∞, . . ., ∞) .Its image, denoted by A(e), is the closure of A(e) and may be naturally identified with We refer to A(e) as a compactified apartment.
Example 1.14.In Example 1.12, every apartment is the union of exactly two one-dimensional cones: a (homothety class of a) seminorm is diagonalizable by a basis {v 1 , v 2 } ⊂ (K 2 ) * if and only if it lies in the cone corresponding to v 1 or v 2 .
Our conventions are chosen so that the associations V → X (V ) and V → B(V ) are covariant functors from finite-dimensional K-vector spaces to topological spaces.In particular, any embedding ι : V ֒→ W induces an embedding B(ι) : B(V ) ֒→ B(W ).We have a natural operation of PGL(V ) on X (V ), which respects diagonalizability and non-degeneracy.It is easy to show, that for dim V > 1 this operation of PGL(V ) on B(V ) is transitive if and only if the valuation on K is surjective.
Example 1.15.In the case of a discretely valued field K, the affine building B r (K) is a flag simplicial complex whose vertices correspond to equivalence classes of lattices (see Section 6.2).Let O K denote the valuation ring, π a uniformiser, and for x ∈ K r+1 , and one recovers the lattice as the closed unit ball of the norm.Given a lattice L, the star of [L] in B r (K) can be identified with the spherical building B r (k) (see Example 1.10) by sending a flag of lattices πL ⊂ L k ⊂ . . .⊂ L to its image in L/πL, which is a flag of k-linear subspaces.
Example 1.16.Let r = 1 and K = Q p .Then the affine Bruhat-Tits building B 1 (Q p ) is an infinite tree whose vertices have valency p + 1.Let e = (e 1 , e 2 ) be a basis of (Q 2 p ) * .The apartment A(e) is an infinite path in the tree which uses all Z p -lattices with basis (p u1 e 1 , p u2 e 2 ) where (u 1 , u 2 ) ∈ Z 2 .See Figure 2 for the compactified Bruhat-Tits tree of Q 2 .The open part is the usual trivalent infinite tree by the circle) can be identified with P 1 (Q 2 ) since any homothety class of a non-trivial proper seminorm on (Q 2 2 ) * can be identified with its kernel, which is a 1-dimensional subspace of (Q 2 2 ) * .

Analytification and tropicalization
For a complete non-Archimedean field K we recall the Berkovich analytification for any locally finite type scheme over K.We describe the relations between the Berkovich space, the space of seminorms on (K r+1 ) * , and the compactified building.This allows us to define a tropicalization map from the space of seminorms on (K r+1 ) * to tropical projective space.
Let A be a finitely generated K-algebra and write U = Spec A.
We think of the set U an of multiplicative seminorms on A as a space, the analytification of U in the sense of Berkovich [Ber90]; we write x ∈ U an for a point in U an as well as |.| x for the associated seminorm.The (analytic) topology of U an is the coarsest that makes all evaluation maps for f ∈ A continuous.For a scheme X that is locally of finite type over K, we define its analytification X an locally as above, and globally by gluing over affine open covers.See [Ber90, Chapter 3] for details.
Remark 2.2.For an affine scheme U of finite type over K, we have a natural inclusion For any scheme X, locally of finite type over K, these inclusions on affine open subsets glue to an inclusion X(K) → X an .If the valuation on K is non-trivial, the image of X(K) is dense in X an for an algebraic closure K of K.
The association X → X an is a covariant functor that commutes with the inclusion of the K-points of a scheme into its analytification.See [Gub13, Section 2.6] for details.
Remark 2.3.To distinguish between multiplicative seminorms on a K-algebra and seminorms on a finite-dimensional K-vector space, we denote the former by |.| x and the latter by ||.|| x .Let V be a vector space over K of dimension r + 1.
Remark 2.4 (Analytification of projective spaces).As explained in [RTW15, Section 2.1.1 ], the analytification of the projective space P(V ) can be identified with the quotient of A(V ) an − {0} modulo homothety.
Let S • V * be the symmetric algebra of the dual vector space V * .This is a finitely generated graded Kalgebra.Every choice of a basis (e 0 , . . ., e r ) of V * induces an isomorphism of S • V * with the polynomial ring over K in r + 1 indeterminates.The affine space A(V ) and projective space P(V ) are defined as As said above, A(V ) an consists of all multiplicative seminorms on S • V * .Then the analytification P(V ) an is the quotient of A(V ) an − {0} by homothety: we define x ∼ y if and only if there exists a constant c > 0 such that for all Here the analytic topology on P(V ) an is equal to the quotient topology.
A multiplicative seminorm on S • V * induces a seminorm on V * = S 1 V * by restriction, hence we have a natural continuous map τ : A(V ) an −→ N (V ) such that τ (x) = 0 if and only if x = 0. Since this map is compatible with the equivalence relations, it descends to a continuous restriction map Proposition 2.5.Let K be spherically complete.Then the restriction map admits a section Given a diagonalizable seminorm ||.||, we may choose a basis e 0 , . . ., e r of V * and c 0 , . . ., c r ∈ R ≥0 such that λ i e i = max The multiplicative seminorm J(||.||) on S • V * is defined by a I e i0 0 . . .e ir r = max When K is a local field, the section J is continuous.
Proof.We refer the reader to [RTW12, Section 3] for details on this construction.Note that in [RTW12] the authors assume that K is local, although everything but the continuity of J goes through when K is spherically complete.
Remark 2.6.We do not know whether J is continuous, when K is not local.Luckily this statement is not needed in the remainder of this article.
Proposition 2.7.Let K be any complete non-Archimedean field.Then the restriction map is surjective.
Proof.We want to construct a section X (V ) → P(V ) an in the general case.We pick any spherically complete extension L/K and denote V ⊗ K L by V L .Then for any seminorm ||.|| on V * we have an induced seminorm ||.|| L on V * L given by ||w|| L = inf max where the infimum runs over all possible decompositions w = i λ i v i with λ i ∈ L and v i ∈ V * .It has been shown in [BE21, Proposition 1.25] that the map X (V ) → X (V L ) given by ||.|| → ||.|| L is injective and that this map is a section of the natural restriction map X (V L ) → X (V ).Now, we define a section of τ by composing where the last map is the restriction of a multiplicative seminorm on the symmetric algebra Corollary 2.8.The space X (V ) is compact.
Remark 2.9.We do not know if the section constructed in the proof of Proposition 2.7 is independent of the choice of the spherical completion (which need not be unique in some cases in positive characteristic, see [BCS18, Theorem 6.17]), or if the section is continuous.
Remark 2.10.We can now show that N (V ) ⊆ N (V ) need not be open.Let K be algebraically closed.The set of K-points of P(V ) is dense in P(V ) an and gets mapped to homothety classes of proper diagonalizable seminorms in X (V ), thus its image is also dense.Consequently, neither inclusion can be open in this case.
From now on we consider the vector space V = K n+1 together with its standard basis e = (e 0 , . . ., e n ) and the associated dual basis e * = (e * 0 , . . ., e * n ) of V * .This identifies A(V ) and P(V ) with A n+1 = Spec K[t 0 , . . ., t n ] and P n = Proj K[t 0 , . . ., t n ] respectively.As explained e.g. in [Pay09], there is a natural continuous tropicalization map that is compatible with the diagonal G m -operation.Therefore, this induces a tropicalization map Tropicalizing the analytification of the big torus in P n yields only elements with finite coordinates: Proposition 2.11.The tropicalization map trop P n factors as (1) where trop X n : X n (K) → TP n is a continuous and surjective map given by associating to a seminorm Proof.The tropicalization map trop X n is well-defined, since, for two seminorms ||.|| x , ||.|| y on V * together with a homothety x ∼ y we have a c > 0 such that ||.|| x = c • ||.|| y and thus It is continuous, since it is given by evaluation maps in each coordinate, and surjective, since the compactified apartment map (a 0 , . . ., a n ) −→ ||.|| e * ,(a0,...,an) induces a continuous section TP n → B n (K) ⊆ X n (K) of trop X n .The factorization (1) follows from the observation that, under the identification S • V * ≃ K[t 0 , . . ., t n ] the linear one-form e * i is naturally identified with the linear polynomial t i .Therefore, we have for all x ∈ (P n ) an and i = 0, . . ., n.This implies Similarly, tropicalizing the non-compactified space of norms, one obtains all finite points: Let Y ⊆ P n be a Zariski-closed subscheme.Then, following [Pay09,Gub13], the tropicalization of Y is defined to be the projection of Y an under trop P n into TP n .Let L be an algebraically closed extension of K with non-trivial absolute value |.| L .By [Gub13, Proposition 3.8] the tropicalization Proposition 2.12.For a linear embedding ι : P r ֒→ P n the tropicalization Trop(P r , ι) = trop P n ι(P r ) an is equal to the projection of X (ι) X r (K) ⊆ X n (K) under trop X n into TP n .
Proof.Since the restriction map τ is surjective, the commutativity of τ implies that X (ι) X r (K) = τ ι an P r ) an .The factorization of the tropicalization map in Propositon 2.11 then yields the claim.
Henceforth, we will define for any linear embedding ι : P r ֒→ P n the composition π ι by Note that π ι is continuous and surjective.A direct computation shows that if ι = f 0 : . . .: f n for f 0 , . . ., f n ∈ (K r+1 ) * we have for all x ∈ X r (K).

Limits of linear tropicalizations
In this section we will set up and prove Theorem A. We first set up a category of linear embeddings such that tropicalization yields a covariant functor into the category of topological spaces.Definition 3.1.Let I be the category of linear embeddings P r ֒→ U ⊆ P n , where U is a torus-invariant open subset of P n , with morphisms given by commutative triangles where U → U ′ is a toric morphism.
Remark 3.2.The codimension of the complement of U ⊆ P n in Definition 3.1 must be at least r + 1.
Thus, for all j = 0, . . ., r, the Chow group of U of codimension j is isomorphic to that of P n .The degree of ι ′ being 1 can be measured by intersecting its image with a generic linear subspace of codimension r.
Since the same is true for ι, we see that hyperplanes must pull back to hyperplanes along U → U ′ , i.e. this morphism is forced to be linear.
Remark 3.3.Allowing the toric morphism to be defined on a smaller torus-invariant open subset instead of the whole projective space endows our index category with many more morphisms, most notably coordinate projections.This makes I into a cofiltered category: indeed, for any two objects ι i : P r ֒→ U i ⊆ P ni , i = 1, 2, we can find a third one dominating both, namely ι : P r ֒→ U ⊆ P N , where N = n 1 + n 2 + 1 and . And for any two morphisms f, g : U 1 → U 2 commuting with ι i as above, we can equalize them by defining U 0 = {f = g} ⊆ U 1 , and noticing that ι 1 factors through U 0 , and the closure of U 0 in P n1 is a linear subspace P n0 ⊆ P n1 .
The tropicalization U trop of a torus-invariant open subset or P n is a special case of the tropicalization of toric varieties, as introduced in [Pay09, Section 3].In our case this means removing those tropical torus-orbits from TP n , for which the corresponding algebraic torus orbit is not contained in U .For a toric morphism ϕ : U → U ′ such that ϕ • ι = ι ′ , we have a natural induced map ϕ trop : U trop → U ′ trop such that ϕ trop Trop(P r , ι) ⊆ Trop(P r , ι ′ ).In particular, if φ is a coordinate projection, then φ trop is the analogous tropical coordinate projection.We refer the reader to [Pay09], to [Rab12, Section 3 and 5], and to [MS15, Section 6.2] for more details on the tropical geometry of toric varieties.
Let ι : P r ֒→ U be a linear closed immersion, where U is a torus-invariant open subset of P n , and let  : U → P n be the inclusion.Then we have a homeomorphism of tropicalizations Using this observation we will henceforth identify any tropicalization arising from a linear morphism P r → U with a subset of TP n .Lemma 3.4.Let ι : P r ֒→ U ⊆ P n and ι ′ : P r ֒→ U ′ ⊆ P n ′ be linear embeddings and ϕ : U → U ′ be a toric morphism with ϕ • ι = ι ′ .Then the diagram X r (K) Trop(P r , ι) Proof.This follows immediately from Proposition 2.11 and the fact that the same holds for the Berkovich analytification [Pay09].
This lemma yields a well-defined continuous map where the limit runs over all linear embeddings ι : P r → U ⊆ P n into torus-invariant open subsets.The limit is endowed with the coarsest topology making all projections continuous; in particular, the limit topology is generated by (the preimage under projection of) all opens in all (finite) tropicalized linear spaces.The right-hand side is thus a pro-object in the category of topological spaces.
Theorem 3.5 (Theorem A).The map Proof.As lim ← −ι∈I Trop P r , ι is a Hausdorff space and X r (K) is compact by Proposition 2.7, it suffices to show bijectivity.
Claim: For all linear embeddings ι = e * 0 : f 1 : • • • : f n , the first coordinate y ι,0 is not ∞.After composing with the corresponding embedding into projective space we can assume that the codomain of ι is P n .We consider the linear embedding e * 0 : • • • : e * r : f 1 : . . .: f n : P r → U ⊂ P r+n where U = P r+n \ V (x 0 , . . ., x r ) ∪ V (x 0 , x r+1 , . . ., x r+n ).Then we have a projection U → P r onto the first r + 1 coordinates and a projection U → P n given by [x 0 : . . .: x r+n ] → [x 0 : x r+1 : . . .: x r+n ].Since (y  ) ∈J is an inverse system, this shows that the first coordinate of y ι cannot be ∞.
We show that this is independent of the choice of .
We check that the constructed map is indeed a seminorm.For f ∈ (K r+1 ) * and λ ∈ K consider any embedding  = e * 0 : f : λf : . . . .Then, by Proposition 2.12, for every y  ∈ Trop(P r , ) there is a class of a seminorm ||.|| ′ ∈ X r (K) with , where val is the valuation on K. Therefore For f, g ∈ (K r+1 ) * , the inequality ||f + g|| ≤ max{||f ||, ||g||} follows similarly by considering an embedding containing f, g and f + g.By construction, the seminorm ||.|| is an inverse image of (y  ) ∈I : Let  = f 1 : . . .: f n : P r −→ U ⊆ P n−1 be a linear embedding into a torus-invariant open subset U ⊆ P n−1 .Consider the embedding  ′ = e * 0 : f 1 : . . .: f n : P r −→ V ⊆ P n where V is the complement of the intersection of the last n coordinate hyperplanes.Since the projection of V onto the last n coordinates is a toric morphism, we have an induced projection map Trop(P r ,  ′ ) −→ Trop(P r , ).The same permutation argument as before shows Remark 3.6.Instead of the category I used above, several subcategories would yield the same limit: (a) The full subcategory I ′ of I of non-degenerate embeddings where no coordinate equals 0. Then I ′ is cofinal in I, and thus the respective limits of tropicalizations are naturally isomorphic.(b) The full subcategory of I of non-degenerate embeddings which are different in every coordinate.(c) The wide subcategory of I, where instead of all (linear) toric morphisms we only allow coordinate projections.Note that these morphisms are the only ones used in the proof of 3.5.
Following the proof of Theorem 3.5 one can similarly show that the restriction of the map lim ← −ι∈I ′ π ι to the non-compactified space X r (K) induces a homeomorphism where n ι is the dimension of projective space that is the codomain of ι.
Remark 3.7.In [GK15] the authors consider a projective limit of tropicalizations with respect to all linear re-embeddings of a fixed affine variety.They, in particular, show that this construction recovers the whole Berkovich analytification in the case of an affine smooth algebraic curve.Theorem A may be thought of as a natural linear-algebraic incarnation of the authors' ideas.

Valuated matroids and tropical linear spaces
In this section, we recall some of the basic definitions and results on valuated matroids in the sense of Dress and Wenzel [DW92], in particular how to associate a (projective) tropical linear space to a valuated matroid and to describe its local structure.

Essentials of Matroids.
A matroid M of rank r is given by an arbitrary set E, called the ground set, and a set I of subsets of E, called independent sets, such that the following axioms are satisfied: (I1) The empty set is independent.(I2) Subsets of independent sets are independent.(I3) If A, B ∈ I and |A| > |B|, then there is a ∈ A such that B ∪ {a} ∈ I. (I4) If A is an inclusion-wise maximal independent set, then |A| = r.
Note that if the ground set E is finite, then axiom (I4) follows from (I3), but if E is infinite then (I4) is a necessary axiom.Clearly these axioms are modeled after linear independence of a set of vectors whose span is r-dimensional; e.g.we get a matroid by considering the linearly independent subsets of a subset E of a vector space K n .The rank of A ⊂ E is the cardinality of a maximal independent in A. The family B(M ) of inclusion-wise maximal sets of I are called the bases of M , and by axiom (I2) they determine I.A circuit C ⊂ E is a minimal dependent set, and a flat is a subset F ⊂ E such that |C \ F | = 1 for all circuits C. A set is a flat if and only if adding any other element to it increases its rank.A loop is an element in E that is contained in no basis; equivalently the singleton with only this element is dependent.
The elements A ∈ E r with v(A) < ∞ form the set of bases of a matroid, called the underlying matroid of v.We explicitly do not require the underlying set E to be finite.This is the tropical variety of points v ∈ R ( E r ) such that for all τ ∈ E r+1 and all σ ∈ E r−1 the expression attains the minimum at least twice.It is straightforward to verify that the basis exchange axiom of valuated matroids is equivalent to the minimum in Equation (2) being attained at least twice for all τ ∈ E r+1 and all σ ∈ E r−1 .If v is a point in the interior Dr(E, r) • = Dr(E, r) ∩ R ( E r ) , then the underlying matroid of v is the uniform matroid on E of rank r, i.e. the bases are all subsets of cardinality r.Valuated matroids with different underlying matroids lie at the boundary.That is, if M is a matroid on E of rank r, we obtain the Dressian with underlying matroid M by intersecting with hyperplanes at infinity: Example 4.3 (Realizable valuated matroids).Let K be a field with a non-Archimedean valuation val : K → R.
(a) Let {f 0 , . . ., f n } be a generating subset of K r+1 .The map defines a valuated matroid or rank r + 1.This follows from the Grassmann-Plücker identity: for all v 0 , . . ., v r , w 0 , . . ., w r ∈ K r+1 .All valuated matroids of this form are called realizable.(b) Extending on (a), the map induced by the permutation-invariant map val • det : K (r+1)×(r+1) → R is a valuated matroid, called the universal realizable matroid.(c) If the valuation on K is trivial, then in (a) we have Thus, the notion of an ordinary matroid that is realizable over a fix trivially valued field corresponds exactly to that of a valuated matroid that is realizable over the same trivially valued field.

4.3.
Tropical linear spaces and matroid polytopes.Valuated matroids v with ground set E and rank r + 1 have an associated polyhedral complex L(v) of pure dimension r in TP E that is connected through codimension 1 and which satisfies intersection-theoretic properties analogous to linear spaces.Hence the L(v) are called tropical linear spaces.We recall the definition, some properties, the associated Matroid polytope, and a regular subdivision induced by v that is dual to L(v).When the matroid is realizable, this recovers the coordinate-wise tropicalization of any linear subspace associated to v.
For the following let E = {0, . . ., n} be the (finite) ground set of a valuated matroid v of rank r + 1 with underlying matroid M .Definition 4.4.The tropical linear space L(v) ⊂ TP n associated to v is the locus of those (u e ) e∈E ∈ TP n such that for any τ ∈ E r+2 the minimum in v(τ \ e}) + u e is attained at least twice.
If some f ∈ E is a loop, by taking τ = B ∪ {f } for any basis B, one can see that u f = ∞ for each (u e ) e∈E ∈ L(v).Thus, adding or deleting loops only yields homeomorphic associated tropical linear spaces.Consequently, for simplicity, we now assume that M has no loops.Now, we give a characterization of matroids in terms of polytopes.We use the following notation for the indicator vector in R E of a set A ⊂ E: Definition 4.5.The matroid polytope P M of a matroid M is the convex hull of The valuated matroid v can be regarded as a height function on the vertices of the polytope P M giving rise to the lifted polytope Γ(v), which is defined to be the convex hull of Projecting the lower facets of Γ(v) back to R n+1 induces a polytopal subdivision D v of P M , called the regular subdivision induced by v.By [YY06, Proposition 2.2] a real-valued function from the vertices of P M is a valuated matroid if and only if all the faces of the induced regular subdivision are matroid polytopes.
A vector u ∈ R n+1 selects a face of the regular subdivision induced by v by taking the convex hull of all vertices e B of P M such that v(B) − u • e B is minimized.Such a face corresponds to the polytope of a matroid, the so-called inital matroid M u of M at u.For a loopless valuated matroid v we have that: The interior of the tropical linear space L(v), i.e. the points with finite coordinates and satysfying Equation (2), equals the set The closure operation only adds points with infinite coordinates.
The condition of M u not having loops is related to the minimum being achieved twice in Equation (2).The set L(v) • has a natural polyhedral structure labelled by the initial matroids, where a cell consists of all the points in L(v) whose associated initial matroid is constant, a given M u .By taking the closure, this polyhedral structure extends to L(v).
4.4.Tropicalized linear spaces.Let ι = f 0 : • • • : f n : P r ֒→ P n with f 0 , . . ., f n ∈ (K r+1 ) * be a linear embedding.Let v be the valuated matroid of rank r + 1 on E = {0, . . ., n} associated to the f 0 , . . ., f n , as in Example 4.3.Note that another representative f ′ 0 , . . ., f ′ n of f 0 : • • • : f n is related by multiplying with a scalar λ = 0. Hence its associated valuated matroid v ′ satisfies that v ′ = v + (r + 1) • val λ.So both v and v ′ define the same tropical linear space L(v) and the same underlying matroid M .Moreover, we have: ).The tropical linear space associated with a realizable valuated matroid coincides with the tropicalization of the corresponding linear embedding: Without loss of generality, we may assume that f i = 0 for all i ∈ {0, . . ., n}.Then Trop(P r , ι) equals the closure in TP n of the non-compactified tropicalization ) by considering the matroid of rank r+1 on E whose bases are the bases B = {b 0 , . . ., b r } of M such that v(B)−u b0 −. ..−u br is minimal.This definition is independent of the choice of representative of u.
In [Rin13] the author defines the notion of a local tropical linear space which can be extended to our compact setting, i.e. for tropical linear spaces in TP n .Definition 4.8.Let B = {b 0 , . . ., b r } be a basis of M .The local tropical linear space Trop(P r , ι) B ⊂ TP n is defined as the closure of the set of vectors u ∈ Trop(ι(P r ) ∩ G n m ) such that M u contains the basis B.
The tropical linear space Trop(P r , ι) is the union of all its local tropical linear spaces Trop(P r , ι) B .
Remark 4.9.In terms of polyhedral subdivisions, the (open part) of the local tropical linear space Trop(P r , ι) B ∩ R n+1 /R½ is a polyhedral complex dual to the faces of the regular subdivision D v that contain the vertex e B .For details, we refer the reader to [Rin13, Corollary 2.5].
4.5.The trivial valuation case.Throughout this subsection, we assume that v is trivially valued, i.e. for all A ∈ E r+1 , we have that v(A) = 0 if and only if A is a basis.This way we may identify v with its underlying matroid M and the subdivision D v of P M from Subsection 4.3 is trivial.So the polyhedral complex of L(v) is a fan, known as the Bergman fan of M (cf.[MS15, Section 4.2] ).
The following theorem refines the polyhedral structure of the Bergman fan, by realizing its support as the order complex associated to the poset of flats of M .Theorem 4.10 ([MS15, Theorem 4.2.6]).Let M be a loopless matroid.The cones e F1 , . . ., e F l R ≥0 + R½ in R n+1 /½ for every chain of flats ∅ ⊆ F 1 ⊆ • • • ⊆ F l form a fan with support L(M ).In particular, L(M ) ∩ R n+1 /½ is homeomorphic to the cone over the order complex of the lattice of flats of M .
With this polyhedral structure we can describe the boundary at infinity of the cones, and see that it differs from an usual coordinate-wise compactification at infinity.Namely, for any cone σ given by a chain of flats ∅ ⊆ F 1 ⊆ • • • ⊆ F l , the closure in TP n is given by e F1 , . . ., e F l R ≥0 + R½.In particular, if the i-th coordinate of a point in σ is infinite, then we consider the minimal j such that i ∈ F j , and we have that the i ′ -th coordinate is also infinity for any i ′ ∈ F j .Thus, there is a single maximal stratum in the boundary of L(v) for each cone, which can alternatively be explained by the fact that the cone structure above triangulates the Bergman fan of M (see [MS15, Section 4.2]).
Example 4.11.We consider the embedding P 2 id − → P 2 .Then the associated matroid has as ground set {e * 0 , e * 1 , e * 2 }, and every subset is independent, i.e. we have the uniform matroid U 3,3 .The Bergman fan of U 3,3 consists of a single cone R 2 .However, there are 6 nontrivial flats, namely all three onedimensional subspaces and three two-dimensional subspaces.In Figure 3 we labelled all one-dimensional cones corresponding to non-trivial flats.given by flats of the uniform matroid U 3,3 .This also represents the compactified apartment in the spherical building B 2 (K).

Faithful linear tropicalization
The goal of this section is to show Theorem B from the introduction.We recall its statement: Theorem 5.1 (Theorem B).Let ι : P r ֒→ P n be a linear closed immersion.Then there is a natural piecewise linear embedding J : Trop(P r , ι) → B r (K) that makes the following diagram commute B r (K) Trop(P r , ι) By a piecewise linear embedding J we mean that we have a finite covering of Trop(P r , ι) by subcomplexes such that the image of each subcomplex lies in an apartment and the restriction of J on each subcomplex is piecewise linear.In particular, π ι induces a piecewise linear homeomorphism between the union of apartments B∈B(M) A(B) and the tropicalized linear subspace Trop(P r , ι).
Choose f 0 , . . ., f n ∈ (K r+1 ) * defining the embedding ι : P r ֒→ P n and let v be the corresponding valuated matroid of rank r + 1 on E = {0, . . ., n} as in Example (4.3) (a).As above, we assume that f i = 0 for all i ∈ E. Let B ∈ B (M ) be a basis of the underlying matroid M .Recall that the compactified apartment A(B) in B r (K) denotes the set of all seminorms on (K r+1 ) * diagonalized by B. It is homeomorphic to TP r : a parametrization TP r ∼ − → A(B) is given by This map is well defined because different projective representatives v ′ = v + λ½ give rise to homothetic seminorms For the proof of Theorem B we need a couple of technical results first.We begin with a valuative version of Cramer's rule.Proof.Label the elements of B as b 0 , . . ., b r , with b 0 equal to our chosen b.Using multilinearity of the determinant and properties of the valuation we find: Proof.If B = E, the statement follows from the definition of tropical projective space.Otherwise, assume there is Since B is a basis, we have that v(B) + u b < ∞.But this is the only finite term in the minimum for Definition 4.4 hence it is not attained twice for τ , which is a contradiction.
For the following statement we interpret [Rin13, Theorem 2.6] in our setting and extend to the compactifications.It gives a piecewise linear homeomorphism between the local tropical linear space Trop(P r , ι) B (as in Definition 4.8) and the compactified apartment A(B).
Proposition 5.4.The map J B sending u ∈ Trop(P r , ι) B to the seminorm • B,uB , where u B = u • e B , is a piecewise linear homeomorphism between Trop(P r , ι) B and A(B).Its inverse is the restriction of π ι .Explicitly, the seminorm Proof.First, we define J B on the open dense subset Trop(P r , ι) B ∩ R n+1 /R½ and show that this gives a piecewise linear homeomorphism to A(B).Clearly, the map π ι is injective on A(B).So we show that π ι • J B is the identity on Trop(P r , ι) B ∩ R n+1 /R½, that is for u in the latter set we show Since − log f b B,uB = u b for b ∈ B, we only need to check the equality for k in E \ B. As u ∈ Trop(P r , ι), the condition in Definition 4.4 for B ∪ {k} says that the minimum is attained at least twice in By subtracting u • e k∪B we see it is equivalent to the minimum being attainted at least twice in Since B is a basis of the initial matroid M u , it minimizes the expression v(σ) − u • e σ over all σ ∈ E r+1 .So the minimum is achieved at i = k and some other i = l.That is, for all i ∈ B we have This also gives the alternative description of π ι .We are left to show that π ι A(B) lies in Trop(P r , ι) B ∩ R n+1 /R½ for which we refer to [Rin13, Theorem 2.6].Hence, J B and the restriction of π ι are piecewise linear inverse homeomorphisms between Trop(P r , ι) B ∩R n+1 /R½ and A(B).We now extend J B naturally to a piecewise linear map Trop(P r , ι) B −→ A(B) by sending u = [u 0 : . . .: u n ] to ||.|| B,uB .This is welldefined if u • e B lies in TP r , namely if there is at least one finite coordinate in u • e B ; this is proven below in Lemma 5.3 and using the fact that Trop(P r , ι) = L(v).Then, π ι • J B is the identity on Trop(P r , ι) B since we have seen it is the identity for a dense subset and Trop(P r , ι) B is Hausdorff.Since A(B) is dense in A(B), the image π ι A(B) is exactly the closure of Trop(P r , ι) B ∩ R n+1 /R½ which is Trop(P r , ι) B .As before, J B • π ι | A(B) = id on A(B) which concludes the proof.
Proof of Theorem B. Recall that Trop(P r , ι) is the union of all its local tropical linear spaces Trop(P r , ι) B where B runs over the bases of the matroid M associated to Trop(P r , ι).Thus, we define J locally as J B and show that the maps J B glue.For A, B ∈ B(M ), we show that J A and J B glue on the open part Trop(P r , ι) A ∩ Trop(P r , ι) B ∩ R n+1 /R½.Choose u = [u 0 : • • • : u n ] in the latter set and, as in Proposition 5.4, write u A , u B for the vectors in R r+1 /R½ with the coordinates of u indexed by A and B, respectively.If • A,uA is homothetic to • B,uB , we get a piecewise linear homeomorphism between Trop(P r , ι) A ∩ Trop(P r , ι) B ∩ R n+1 /R½ and A(A) ∩ A(B).
Claim: The seminorm • A,uA is homothetic to • B,uB .First assume that A and B differ by two elements, i.e. there are a ∈ A and b ∈ B such that A\a = B \b. Choose v ∈ (K r+1 ) * and write it in terms of the bases A and B: j∈B\b (α a β j + α j )f j .We want to show equality of the expressions By Lemma 5.2 we have for every j ∈ B that It remains to show for i ∈ A\a that either val(α and moreover if val(α a )+ val(β i ) > val(α i ), then val(α a β i + α i ) = val(α i ) and we are done.Thus, assume that val(α a ) + val(β i ) ≤ val(α i ).In that case, we calculate using Equations (4) and (6), the following Thus, both minima in Equation (3) coincide.For the general case where A∆B has 2m elements, since both A and B are bases in the initial matroid M u , by the basis exchange axiom there is a sequence of bases B 0 , B 1 , . . ., B m such that B 0 = A and B m = B and every pair B q , B q+1 differs by two elements.Thus, we may apply our previous argument to every pair B q , B q+1 to conclude the claim for inner points of the local maps.Again, extending to the compactifications concludes the proof.
Passing to the smaller set U v allows us to endow it with a limit topology as follows.Let I be the category of finite subsets E ′ of E containing a basis, with inclusions as morphisms.Then we have a functor from I op into the category of topological spaces, assigning to each E ′ the space U v ′ ⊆ TP E ′ , where v ′ is the restriction of v to E ′ , and to every inclusion the corresponding coordinate projection.Note that these coordinate projections are well-defined by the construction of the U v ′ .We see that and we can endow it with the limit topology.Since we also have an identification and in particular we can endow L(v) with the limit topology.
We now restrict our attention to realizable valuated matroids.Recall that the universal realizable matroid w univ is given by w univ (A) = val det(A) for A ∈ E r+1 .Theorem A and Speyer's result on the tropicalization of finite linear subspaces (Theorem 4.7) allow us to identify the space of seminorms on (K r+1 ) * up to homothety with the tropical linear space associated to the universal realizable matroid.
Theorem 7.2 (Theorem C).The Goldman-Iwahori space is the tropical linear space associated to the universal realizable matroid w univ , i.e.
Proof.By Theorem A and Remark 3.6 (c), we can write X r (K) as the limit of all linear tropicalizations with respect to the category of coordinate projections.The tropicalization functor from this category is naturally equivalent to the functor which associates to an embedding ι = [f 0 : . . .: f n ] the tropical linear space associated to the valuated matroid given by {f 0 , . . ., f n }, as repeating entries and permuting coordinates yields homeomorphic tropicalizations.
Let E denote the set of (non-zero) vectors in K r+1 .We obtain linear maps (7) E K −։ K r+1 , and dually As in Example 4.3 and Section 4.4, we may associate to ι univ the realizable valuated matroid w univ .Hence, we can interpret X r (K) as the tropicalization of the universal projective linear subspace of rank r.
In the following we will show that X r (K) is cut out by much simpler equations than the ones coming from the universal realizable valuated matroid.We can think of (7) as the (r + 1) × E matrix whose e-th column vector represents e in the standard basis of K r+1 .It follows that the i-th row corresponds to the i-th coordinate projection as a function on E, that is: Proposition 7.3.The image of ι univ : K r+1 ֒→ K E consists of (the restrictions of ) all the linear maps from K r+1 (resp.E) to K. In particular, the equations of ι univ involve only finitely many variables.
Remark 7.4.These are the equations of K r+1 ֒→ K E ′ for any subset E ′ ⊆ E and the corresponding projection K E → K E ′ (restriction of functions).
As a curiosity, we note that the large circuits of Definition 4.4 are equivalent to the tropicalization of the small circuits from Proposition 7.3.A tight span is an isometric embedding of a metric space E into a hyperconvex metric space T E .The motivation for these spaces is fitting phylogenetic data; see the cited work for a discussion.There are also applications to extending valuations in p-adic geometry [DT93].A so-called four-point condition [DMT96, Section 4.6] is necessary and sufficient for a tight span to be an R-tree; see Figure 1 for an example of an R-tree.This condition is essentially the basis exchange property for rank-2 valuated matroids.Hence, generalizing this to higher dimensions, [DT93] introduces the tight span of a rank-r valuated matroid (E, v) as The maximum in Equation (12) says that the functions p efficiently satisfy a triangle inequality.Their formulation using max is dual to our work using min.
In parallel to Proposition 5.4, there is a local description of T (E,v) indexed by bases.Given a basis B = {b 1 , . . ., b r }, the map Φ B that sends a point (u 1 , . . ., u r ) in the hyperplane H v(B) = is injective [DT98, Proposition 1].There is a similar polyhedral description for intersections Φ A (H) ∩ Φ B (H) with A and B bases of (E, v).Moreover, as B varies over all bases of (E, v), the whole T (E,v) is covered.It can be shown via the theory of (B, N ) pairs that the Φ B (H) form the apartments of a building:

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 Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Sachbeihilfe From Riemann surfaces to tropical curves (and back again), project number 456557832, and from the LOEWE grant Uniformized Structures in Algebra and Geometry.L.B. has received funding from the ERC Advanced Grant SYZYGY of the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 834172).A.V. has received funding from the Swiss National Science Foundation, grant number 200142.
rise to coordinates d i via the formula above by setting d l = 1, and thus we obtain a well-defined diagonalizable seminorm with ||.|| Vi\Vi−1 = d i and generic value 1.By construction, these maps are inverses of each other.Remark 1.11.The bijection in Example 1.10 restricts to a bijection

4. 2 .
Valuated matroids.We begin with the following definition due to Dress and Wenzel [DW92].Definition 4.1.A valuated matroid of rank r on a ground set E is a function v : E r −→ R that fulfils the following axioms: (i) There exists a subset A ∈ E r with v(A) = ∞.(ii) For all A, B ∈ E r and a ∈ A − B we have the valuated basis exchange property

Remark 4. 2 .
Valuated matroids on E of rank r are parameterized by the Dressian Dr(E, r) ⊂ R ( E r ) .

Figure 3 .
Figure3.The compactified cones of Trop(P 2 , id) = TP 2 given by flats of the uniform matroid U 3,3 .This also represents the compactified apartment in the spherical building B 2 (K).

Lemma 5. 2 .
Let B be a basis of M , andk ∈ E \ B. Write f k = b∈B λ b f b with λ b ∈ K.For all b ∈ B we have val(λ b ) = v k ∪ B \ b − v(B).

)
As u ∈ Trop(P r , ι) B , the basis B is in the initial matroid M u , so B minimizes the expression v(σ)−u•e σ over all σ ∈ E r+1 , so v(B)− u • e B ≤ v(A ∪ B \ j) − u • e A∪B\j for all j ∈ B. By Equation (4) we get u a − u j = u • (e A∪B\j − e B ) ≤ v(A ∪ B \ j) − v(B) = val(β j ).(5) Also u is in Trop(P r , ι) A , so A is in the initial matroid M u , thus v(A) − u • e A ≤ v(B) − u • e B and so u a − u b ≥ v(A) − v(B) = val(β a ).(6)If we set j equals b in Equation (5) and combine it with Equation (6) we get val(β b ) + u b = val(β a ) + u a , and furthermore val(α a ) + val(β b ) + u b = val(α a ) + u a .So equality in the first terms of Equation (3) happens.

r
i=0 u i = v(B) to the linear map Φ B (u) : E → R given by e −→ max i∈B v(e ∪ B \ i) + u i − v(B) Example 1.2.Pick a basis e = (e 1 , . .., e n ) of V and a = (a 1 , . .., a n ) ∈ R n .We may associate to this datum a seminorm ||.|| e, a on V given by associating to v = n i=1 λ i e i the value max i=1,...,n |λ i |e −ai .The seminorm is non-trivial if and only if at least one a i = ∞ and it is a norm if and only if all a i = ∞.A seminorm of the form ||.|| e, a for a basis e and coefficients a ∈ R Definition 1.1.A norm on V is a map ||.|| : V → R that fulfills the following axioms: (i) For all v ∈ V we have ||v|| ≥ 0 and ||v|| = 0 if and only if v = 0; (ii) For all v ∈ V and λ ∈ K we have ||λ • v|| = |λ| • ||v|| .(iii)For all v, w ∈ V the strong triangle inequality ||v + w|| ≤ max ||v||, ||w|| holds.If in (i) we only require ||v|| ≥ 0 and allow vectors v ∈ V − {0} with ||v|| = 0 we say that ||.|| is a seminorm.A seminorm ||.|| is said to be non-trivial if there is a v ∈ V such that ||v|| = 0. n is said to be diagonalizable.Definition 1.3.A non-Archimedean field K is said to be spherically complete if any decreasing sequence of closed balls has non-empty intersection.