Presentations of Transversal Valuated Matroids

Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Pl\"ucker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint of matroid theory, the tropical Stiefel map defines a generalization of transversal matroids in the valuated context, and our results are the valuated generalizations of theorems of Brualdi and Dinolt, Mason and others on the set of all set families that present a given transversal matroid. We show that a connected valuated matroid is transversal if and only if all of its connected initial matroids are. The duals of our results describe complete stable intersections via valuated strict gammoids.


Introduction
In tropical mathematics, the accepted definition of tropical linear spaces uses an analogue to vectors of Plücker coordinates. These vectors were introduced by Dress and Wenzel [16], who named them valuated matroids because matroids appear as a special case.
Over a field K, every linear subspace of K n can also be described as the rowspace of some matrix with entries in K. The tropical counterpart fails. The tropical Stiefel map π of [20] sends a matrix of tropical numbers to the tropical linear space determined by its vector of maximal minors; however, not all tropical linear spaces arise in this way.
The combinatorics of the map π is governed by transversal matroids. Let A = { {A 1 , . . . , A d } } be a multiset of subsets of a finite set E. Edmonds and Fulkerson [18] observed that the set of subsets J ⊆ E which form a transversal of A, i.e. such that there is an injection f : J → {1, . . . , d} with j ∈ A f (j) for each j ∈ J, are the independent sets of a matroid. A matroid M arising in this way is called a transversal matroid, and A is called a presentation of M . To emphasize the commonality between valuated and unvaluated cases, we define a transversal valuated matroid V to be a valuated matroid in the image of π, i.e. a vector of tropical maximal minors of a d × n matrix A of tropical numbers. The matroids that are transversal valuated matroids are exactly the transversal matroids. Brualdi  This directly generalizes Brualdi and Dinolt's result to valuated matroids. For (unvaluated) matroids in the image of π, the apex of our fan is the unique maximal presentation of Brualdi and Dinolt. Apart from a lineality space, all rays of our fan are in coordinate directions, and the sets of coordinates that appear are described by a "local" reformulation of equation (1.1).
In [20] a necessary condition for a valuated matroid V to be transversal was given (Proposition 3.6). Assuming for convenience that V is connected, the condition is that if V is transversal, all connected initial matroids of V must be transversal. The initial matroids are those whose matroid polytopes appear in the polytope subdivision induced by V . We obtain a converse.

Theorem 1.2 (= Theorem 6.20). A connected valuated matroid is transversal if and only if all of its connected initial matroids are transversal.
Duality of valuated matroids replaces the tropical Stiefel map by the process of taking the stable intersection of a collection of tropical hyperplanes. In the realm of matroids, the dual of the class of transversal matroids is the class of strict gammoids. This class arises from flows in directed graphs, which admit a natural generalization to the realm of valuated matroids which we call valuated strict gammoids. We find the statements derived from Theorem 6.20 by this duality to be of interest in their own right. Theorem 1.3 (= Theorem 7.5). Let V be a valuated matroid and L its corresponding tropical linear space. Then the following are equivalent: (1) L is the stable intersection of tropical hyperplanes.
(3) Near each point, L is locally the Bergman fan of a strict gammoid. Furthermore, Theorem 6.6 explicitly describes the spaces of all d-tuples of tropical hyperplanes whose stable intersection is a given tropical linear space, and of all weighted directed graphs that present a given valuated strict gammoid.
In this paper, Section 2 reviews valuated matroids and tropical linear spaces. Section 3 introduces transversality and the Stiefel map, and interprets the former as the {0, ∞}-valued case of the latter. We begin to characterize presentations in Section 4, by bounds on the number of rows chosen from certain regions of the tropical linear space. Section 5 introduces a piece of technical apparatus needed for the proofs of the main theorems, after which Section 6 proves them. Section 7 introduces strict gammoids and stable intersection and reframes our results in this language.
Fix a set [n] = {1, . . . , n}. We denote the set of all subsets of [n] with cardinality d by [n] d . Given a subset J ⊆ [n], we denote its zero-one indicator vector by e J = j∈J e j ∈ R n .
In the theory of valuated matroids, coordinates are drawn from the semiring T = R ∪ {∞} of tropical numbers, with operations ⊕ := min and := + and identity elements ∞ and 0. The set T n of vectors of n tropical numbers plays the role of affine n-space in tropical geometry. But we prefer to work in projective space: 1) where the action of R(1, . . . , 1) is by addition. When we speak of the relative interior relint(P ) of a polyhedron P ⊆ TP n−1 , we exclude the points which have more coordinates equal to ∞ than a generic point of P does, i.e. the points on the "faces at infinity" of P .

Valuated matroids and matroid polytopes. A valuated matroid
V on the ground set [n], whose rank is an integer rk(V ) = d with 0 ≤ d ≤ n, is a vector in TP ( n d )−1 whose coordinates are labeled by and C ∈ [n] d+1 , there is more than one index j ∈ C \ A at which V A∪{j} + V C\{j} attains its minimal value.
Given a valuated matroid V , the set of all B ∈ n d such that V B is finite is the set of bases of a matroid, called the matroid underlying V . Following the notation used in [5], we write V for the matroid underlying V . A vector x ∈ R n selects a face of the regular subdivision induced by V by taking the convex hull of all vertices e B of P B such that V B − i∈B x i is minimized. Such a face corresponds to the polytope of a matroid which we write V x known as the initial matroid of V at x. We write M(V ) for the set of all initial matroids of V all of whose loops are loops in V . Example 2.1. Consider the uniform matroid U 2,4 . Its matroid polytope is the hypersimplex ∆ 2,4 which is an octahedron. Now consider the valuated matroid V where V 34 = 1 and V B = 0 for every B ∈ 4 2 \{34}. The matroid subdivision induced by V divides the octahedron into two square pyramids, one with apex e 12 and the other one with apex e 34 . The only x that selects the pyramid with apex e 12 is [0 : 0 : 0 : 0] while the only x that selects the pyramid with apex e 34 is [0 : 0 : 1 : 1]. The initial matroids contained in M(V ) are those whose polytopes are the two square pyramids, their common square face, and four of the triangular faces, namely conv{e 12 , e 13 , e 14 } and its S 4 -images.
2.2. Tropical linear spaces. The (projective) tropical linear space associated to a valuated matroid V is d+1 , more than one j ∈ C minimizes x j + V C\{j} .} We call L(V ) a tropical hyperplane if V has rank d = n − 1.
We describe the polyhedral structure of a tropical linear space L = L(V ) using the language of matroids. For simplicity, we assume throughout that V has no loops or coloops. Define We have that L is the closure of L • /R(1, . . . , 1) within TP n−1 , where the closure operation only adds points with infinite coordinates ( [45,Prop 2.3]; implicit in [28]). The complex L is pure of dimension d − 1. The polyhedral complex structure of L is determined by the faces in L • : the interiors of these faces are the sets of points x ∈ R n such that the matroid V x is constant. For a matroid M ∈ M(V ), we write L M for its corresponding cell, that is: where J is the set of all nonloops of M and ι J : R J → T n is the inclusion filling in infinities in the missing coordinates. When this cell is 0-dimensional, i.e. when M is connected, we call it v L M (pedantically, v L M is the point which is the single element of L M ). Example 2.2. Consider the valuated matroid V from Example 2.1. The polytopes in the subdivision induced by V that correspond to loopless matroids are the two square pyramids, the square separating the pyramids and the four triangles which are inside each of the hyperplanes x i = 1 for i ∈ [4]. Figure 1 shows a picture of the associated linear space.
If M is a matroid, the polyhedral complex structure we have just placed on the tropical linear space L(M ) is the Bergman fan as in [19], with the 'coarse subdivision' as in [4].
We will use a construction of the set L(M ) in terms of flats throughout.
The above shows that, as a set, the Bergman fan is the order complex of the lattice of flats, which endows the Bergman fan with its 'fine subdivision' structure, also known as the nested set complex of M .
If L = L(V ) is a tropical linear space and x ∈ R n /R(1, . . . , 1) is in the relative interior of L M , then L(M ) equals the set of vectors y such that x + εy ∈ L for all sufficiently small ε > 0. That is, L looks like the translation L(M ) + x locally near x.
Valuated matroids have analogs of dual, restriction and contraction. The dual of V is the valuated matroid V * of rank n − d given by Let J be an arbitrary subset of [n] and B c any basis of M/J. Then the restriction of V to J is the valuated matroid V |J on the ground set J of rank k = d − |B c | such that V |J B = V B∪Bc for any B ∈ J k . This definition does not depend on the choice of B c ∈ B(M/J), as choosing a different basis means tropically scaling all Plücker coordinates by the same factor. In Lemma 4.1.11 of [21] describes the effects of deletion and contraction on L(V ). Given a subset A ⊆ [n] we have that

Transversality
We recommend [9] as a general reference for transversal matroids.
3.1. The tropical Stiefel map. The fibers of the following map π are our main subject.
The minimum on the right hand side of this equation, over the d! allocations of the names j 1 , . . . , j d to the elements of B, is a tropical maximal minor of A. The history of the connection between transversals and determinants goes back at least to [17].
Remark 3.2. The domain of π is the subset of T d×n where at least one injective function j : Hall's theorem, the only matrices excluded from the domain are those that have a k × (n + 1 − k) submatrix all of whose entries are ∞ for some 1 ≤ k ≤ d.   34 = 1, which is the same valuated matroid as in Examples 2.1 and 2.2. Notice that replacing either A 1,1 or A 1,2 (but not both at the same time) by any tropical number larger than 0 does not change any of the minors, so the resulting matrix would be mapped to the same valuated matroid. Similarly, replacing either A 2,3 or A 2,4 by a number larger than 1 also does not change π(A). Figure 1 shows the tropical linear space of π(A). Any matrix A with π(A ) = π(A) must have one row giving projective coordinates for a point in the blue subcomplex of the figure, and the other row doing the same for the red subcomplex. Later, we will show how all fibers of π have a similar behavior.
Permuting the rows of A, or adding a scalar to any row, does not change π(A), and therefore neither does left multiplication by any invertible tropical matrix. The first invariance implies that π(A) is determined by the list of the projectivization (lying in TP n−1 ) of each row of A, and the second invariance means that π(A) is determined by the unordered list, i.e. the multiset, of these projectivizations. So we will normally discuss fibers of π in terms of such multisets. where A is a matrix whose rows are coordinate vectors for the elements of A.
If we say that a multiset A is a presentation of a tropical linear space L(V ), we mean that it is a presentation of V .
The tropical Stiefel map is not surjective onto the space of valuated matroids. In [20] the name Stiefel tropical linear space was given to tropical linear spaces of the form L(π(A)). We grant the valuated matroids another name motivated in what follows: Note that a transversal valuated matroid is not merely an arbitrary valuated matroid whose underlying matroid is transversal. A counterexample is the valuated matroid V of Figure 2, whose underlying matroid is the transversal matroid U 2,6 , but which is not transversal itself as explained in Example 3.10.
Let us understand why Definition 3.5 agrees with the classical definition of a transversal matroid. Classically, a set system presentation of a transversal matroid on [n] is a multiset A of subsets of [n]. A set is independent if there is a matching i.e. J is independent if there is an injective function σ : J → A such that j ∈ σ(j) for every j ∈ J. Such a set system presentation A can be turned into a presentation in our sense by replacing each element [n] \ F ∈ A by e F where In the corresponding {0, ∞}-matrix A, we have that π(A) B = 0 if there is matching from B and ∞ otherwise. Conversely, given a transversal valuated matroid V = π(A), the multiset consisting of the set of finite entries of each row of A is a presentation of V . We caution readers of the literature on transversal matroids that most authors allow the set system presenting a rank d matroid to contain more than d sets. These authors would say that all our presentations are "of rank d".
Here is a necessary condition for transversality of valuated matroids. In Theorem 6.20 we show that this condition is also sufficient.   At the heart of this paper is the idea of generalizing the above result to valuated matroids.
The literature contains several statements similar or equivalent to the above. Below we describe another reformulation of Proposition 3.7 as a precise bijection between integer vectors and presentations. See Bonin [7] for more detail on the equivalence.
Notice that if M is a transversal matroid, extending τ to be 0 for every non-cyclic flat yields a solution of the integer program in Proposition 3.8. This is the minimal such function in the following sense: if β is a solution of this system for some matroid M , then by Proposition 3.7 we have that for every F ∈ CF(M ) Testing if M is transversal can be done by checking whether τ (as defined in Equation (3.2)) satisfies inequalities (3.3) and (3.4). Another test for transversality, Proposition 6.8, was provided by Mason and Ingleton.
The above discussion shows that every set system presentation of M can be obtained from the maximal presentation by replacing some elements F with G where cocl(G) = F . Therefore, every set system presentation of M is obtained from the maximal presentation by adding relative coloops to the flats chosen. The non-cyclic flats of U d,n are all sets F such that 0 < |F | < d. Inequality (3.4) says that for any J ⊆ [n] with |J| < d, there cannot be more than d − |J| sets among the complements of a presentation of U d,n that are supersets of or equal to J. Because a proper flat of U d,n has at most d − 1 elements, the case |J| = d of the last sentence is true as well. Proposition 3.8 says that any set system of d sets satisfying these conditions is a presentation of U d,n . After translating to matrices via equation (3.1), this is the statement (c)⇔(d) of [20,Proposition 8]. The reader may check that when n = d one recovers Philip Hall's marriage theorem, and when n = d + 1, the dragon marriage theorem of Postnikov [38].  Similar reasoning shows that no rank 2 matroid with three or more nontrivial parallel classes has a transversal presentation. The nontransversality of a valuated matroid can be seen in the geometry of the corresponding linear space. For example, the tropical linear space in Figure 2 has a vertex incident to 3 bounded edges. This vertex corresponds to the non-transversal matroid M and each bounded edge corresponds to one of its non trivial cyclic flats. This provides one proof that the tree formed by the bounded faces of a Stiefel tropical linear space of rank 2 is a path.

Additional remarks.
Remark 3.11. The image of π is always contained in the tropical Grassmannian TropGr(d, n), the tropicalization of the Grassmannian over a field in its Plücker embedding [43]. The matroid of Example 3.10 lies in the tropical Grassmannian for any field, so π does not surject onto TropGr(d, n). Remark 3.12. A family of presentations that have been the focus of much previous work are the pointed presentations, where A has a tropical identity matrix as a maximal submatrix [24,27,41]. The unvaluated matroids with pointed presentations are called fundamental transversal matroids [7, Section 3.1] (see also [6,39]); by Proposition 4.8, these presentations can be taken to be by {0, ∞} matrices. If V has a pointed presentation A, then all facets of P V share the vertex e J where A J is the identity submatrix. The converse is false: for example, nonfundamental transversal matroids exist, and for these P V has only one facet. In other words, whereas the Grassmannian Gr(d, K n ) over a field K has an atlas of charts isomorphic to A d(n−d) K , one for each position of the identity submatrix, the corresponding maps from T d(n−d) fail even to cover the image of π.
Remark 3.13. If V and V are valuated matroids on [n] of respective ranks d and d , their stable sum V + V is the valuated matroid of rank d + d defined by Stable sum generalizes matroid union in the special case that the matroid union is additive in rank, for which reason Frenk [21, Section 4.1] calls it the "valuated matroid union". In this language, presentations are decompositions of a valuated matroid as a stable sum of rank 1 valuated matroids.
Remark 3.14. A way of looking at the tropical Stiefel map which we do not take up here is in terms of the semimodule theory of T. This viewpoint is adopted in [12], and is generalized in [34] to the valuated version of Perfect's "induction" of a matroid across a directed graph [37].

Characterizing presentations by regions
In this section, we characterize presentations of a valuated matroid V in terms of bounds on the number of points which may lie in certain regions of L(V ).
We start by noting that the search for transversal presentations of a tropical linear space L is helpfully delimited by the fact that all elements of a presentation must lie in L. This is essentially the tropical Cramer rule [2,40], but the proof is short so we include it for convenience.
Proof. Write the presentation as a matrix A ∈ T d×n . Define an expanded matrix A (i) whose first d rows agree with A and whose (d + 1)st row equals its ith row. Given a set C ∈ By construction of A (i) , swapping the ith and (d + 1)th entries of the transversal preserves this sum. This implies that both k = j(i) and k = j(d + 1) minimize the quantity A i,k + L C\{k} , because in each case L C\{k} is the sum of the matrix entries in the transversal other than the entry in the (d + 1)th row, which contributes A i,k . Therefore the tropical equations in the definition of L(V ) hold at A i . Our next step is to generalize Proposition 3.8, which characterizes set system presentations of matroids, to describe presentations of unvaluated matroids by points with unrestricted tropical coordinates. In this case, the regions we invoke can be seen as generalizing the ranges of summation in inequalities (3.4) and (3.5).
For that purpose we define relative support. This is essentially the same notion as covectors in the theory of tropical hyperplane arrangements [3, Section 3]. The covector of a point is the list of complements of its relative supports with respect to the apex of each tropical hyperplane.

Definition 4.2.
Let x and y be two points in TP n−1 such that x has finite coordinates. The relative support rs x (y) ⊆ [n] of y with respect to x is the set indexing the coordinates where y − x does not attain its minimum.
Note that addition of a scalar multiple of (1, . . . , 1) to the coordinates of a point does not affect its relative support, so the relative support is well defined. If x has a fixed vector of affine coordinates (x 1 , . . . , x n ) ∈ R n , then we say that the supportive choice of affine coordinates (y 1 , . . . , y n ) for y, with respect to (x 1 , . . . , x n ), is the one which achieves min j (y j − x j ) = 0. In terms of supportive coordinates, Definition 4.2 becomes . By definition of L, we have that rs 0 (y) ∈ F(M ) for every y ∈ L. So for each flat F ∈ F(M ) we define the region In supportive coordinates with respect to the zero vector, R 0 (F, L) consists of all the points which have positive entries in the coordinates indexed by F . Similarly, for each cyclic flat F ∈ CF(M ) we define another region In other words, R 0 (F, L) consists of all points y in L where no coordinate of y in F achieves the minimum among its coordinates and R ∞ (F, L) are those points in L whose coordinates in F are ∞.
where F is a flat in the first line, and a cyclic flat in the second.

presentation of M if and only if the following conditions hold:
( Proof. Let A ∈ T d×n be the matrix whose rows are the supportive coordinates for A 1 , . . . , A d with respect to 0, so all entries are nonnegative and each row contains a zero. First we assume that There are k + 1 rows with positive coordinates in all of the columns indexed by F . This means that in the square d × d submatrix given by the columns of B, there is a (k + 1) × (d − k) submatrix whose entries are all positive. Then the tropical minor corresponding to B must be positive, which is a contradiction as M B = 0. Now suppose there is a cyclic flat F ∈ CF(M ) that violates condition (2). As we already proved condition (1) is satisfied, we can assume σ ∞ (A, F ) < cork(F ) = k. Then there are d − k + 1 rows with finite entries in the columns corresponding to F . Assume there is a matching of the submatrix of F with these rows. Then any matching of the whole matrix can be used to get a matching that uses the columns of F in all of those d − k + 1 rows by exchanging the entries. This is a contradiction to the rank of F ; so no such matching exists, and there must be a violation of Hall's condition. Let I be the violating subset of rows of size m, so that there are at most m − 1 columns with which elements of I can be matched. Let j be one of those columns. Because F is cyclic there should be a matching of d − k rows to F − j. So there is a row i corresponding to a point in R ∞ (F, L) which is not used in this matching. Then I − i has access to at most ≤ m − 2 columns of F − j, which is a contradiction to the matching.
We now do the other direction. Assume conditions (1) and (2) are satisfied. Because A i ∈ L, we have rs 0 (A i ) ∈ F(M ). Consider the initial matroid M = π(A) 0 , that is, the matroid whose bases are given by the entries where π(A) is 0. This M is transversal, and Condition (1) implies that all independent sets in M are also independent sets in M (see Lemma 4.4 in [10]). This means that for each B ∈ B(M ) there is a matching on the 0 entries of A, so that B ∈ M . (2) there are d − k rows with infinity entries at the columns of F . This means that in the square We now turn our attention to the more general case L = L(V ) where V is any valuated matroid. When we look at general tropical linear spaces, we have to define the regions R 0 and R ∞ more carefully. They will now have three parameters: the tropical linear space L = L(V ), a point x ∈ L with finite coordinates and a flat F ∈ F(M ) such that the relative interior of L M contains x. Before we define these regions, we provide the following lemma which explains why it still makes sense to take flats as parameters. Proof. Notice that x being in the relative interior of L M already implies that x has finite coordinates, so it makes sense to talk about rs x (y). Without loss of generality we can translate L so that x is the origin. In this case, we may assume that V B = 0 if and only if B ∈ B(M ). Now suppose that there exists y ∈ L such that rs x (y) / ∈ F(M ). This means there is an element ∈ rs x (y) then V B > 0. So the minimum is only attained once, which is a contradiction.
Given a tropical linear space L = L(V ), a matroid M ∈ M(V ), a flat F ∈ F(M ) and a point x ∈ relint(L M ), we define two regions, which we will use to constrain the possible position of points in presentations.
See Example 4.13 for examples of these definitions.

Lemma 4.5. Let M be a matroid. Then
Proof. The first equivalence is straight forward from the definitions of R 0 (F, 0, L(M )) and R 0 (F, L(M )). To see that R ∞ (F, 0, L(M )) = R ∞ (F, L(M )) note that relint L(M ) M |F ⊕M/F ⊆ R 0 (F, L(M )), so every y ∈ relint L(M ) M |F ⊕M/F has positive entries in F when written in supportive coordinates with respect to 0. Any z ∈ R 0 (F, y, L(M )) must have coordinates larger than y in F when written in supportive coordinates with respect to the 0. As relint L(M ) M |F ⊕M/F is an open cone, y can have arbitrarily large coordinates in F and any z ∈ R ∞ (F, 0, L(M )) must have infinite entries at F , so R ∞ (F, 0, L(M )) ⊆ R ∞ (F, L(M )). But clearly also R ∞ (F, L(M )) ⊇ R 0 (F, y, L(M )) for every y ∈ relint L(M ) M |F ⊕M/F , so the equality holds.
where F is a flat of M in the first line, and a cyclic flat of M in the second. The following lemma shows that Proof. If y ∈ relint(L M |F ⊕M/F ), then y is of the form v L M + c 1 e F 1 + · · · + c k e F k for a flag F 1 ⊂ · · · ⊂ F k containing F and such that 0 ≤ c i < ∞ for every i; the c i are finite because we have excluded faces at infinity from the relative interior. This is the same form as points have in the cone L(M ) F of the Bergman fan of M . This means in particular that for any j / ∈ F and j ∈ F we have y j ≤ y j when written in the supportive coordinates with respect to (fixed coordinates for The following definition helps us use the Bergman fan case for the more general setting of tropical linear spaces. We think of Z x as 'zooming' into x, pushing all points of L away from x to infinity in a straight line. Thus, Z x (L) keeps only local information of L around x.
The corresponding arguments in [20] are Propositions 5.5 and 5.9.
Proof. Let A ∈ T d×n be the matrix whose ith row consists of A i written in supportive coordinates with respect to x. Notice that the scaling of rows in the matrix A does not change π(A) and adding the vector x to each of the rows of A, changes π(A) B by adding j∈B x j . This implies that y ∈ L(π(A)) if and only if y + x ∈ L. So we have that L(π(A)) equals L − x, the tropical linear space L translated so that x is at the origin.
Tropically exponentiating (i.e. classically multiplying) each entry of A by t transforms L − x by a classical homothety centered at the origin of factor t, so L(π(A t )) = t(L − x). When t → ∞, we have that Since tropical linear spaces are locally fans, we have that as t → ∞, t(L − x) goes to the fan with which L − x coincides near the origin. This is the same fan whose translation by x coincides with L near x, namely L(M ), since x ∈ relint(L M ). Because π is a continuous map in its domain, these two limits imply that π(Z x (A)) = L(M ) as long as Z x (A) is still in the domain of π. So the only thing left to prove is that this is the case, namely, that there is a set B for which π B (A) = 0.
If there were no maximal minor of A equal to 0, then there would be an a × b submatrix A of A consisting of strictly positive entries such that a + b > n. Among such matrices A select one where b is maximal, i.e. with the most columns. Let I be the set of rows taken by A and J be the set of columns not taken by A . Notice that |I| = a > n−b = |J|. Consider a bipartite graph G whose vertices are I J and containing the edge (i, j) just if A i,j = 0. If G is disconnected, then there is a connected component with vertices I ⊆ I and J ⊆ J with |I | > |J |. So the submatrix of A given by rows I and columns [n] \ J is strictly positive and has more columns than A , which is a contradiction. So G is connected.
Let j ∈ J. As M has no coloops, then there is a basis B ∈ B(M ) such that j / ∈ B. Because 0 ∈ L(π(A)) M , then π(A) B is minimal among all maximal minors of A. The value of π(A) B is achieved by a matching σ : B → [d]. All matching must use an entry of A , because a + b > n implies that the total number of columns and rows of A not included in A is less than d. So there is an element j ∈ [n] \ J such that σ(j ) ∈ I. Let G be the graph where you add to G the vertex j and the edge (σ(j ), j ). As G is connected, then there is a path G from j to j. The matching given by σ does not use consecutive edges. By replacing each edge used by σ in G by the edge that follows it, we get a matching σ from B − i ∪ j to [d]. But the weight of this matching is less than that of σ as we replaced a strictly positive entry A σ(j ),j by zero. This contradicts the minimality of π(A) B . It is straightforward to check that the collection of flats { {45, ∅, 145} } satisfy the conditions of Proposition 3.7, so their complements are a set system presentation of V x . In other words, the rows of the matrix We will need the following lemma.
By definition of the zoom map Z x , this happens if and only if i ∈ rs x (y) for every i ∈ F , which is equivalent to y ∈ R 0 (F, x, L).   (2) is not satisfied, it means that Then there exists y ∈ L M |F ⊕M/F such that A i / ∈ R 0 (F, y, L). From F ∈ CF(M ) we see that M |F ⊕M/F is coloop-free and F ∈ CF(M |F ⊕ F ), so by Proposition 4.11 we have that cork M/F ⊕M |F (F ) = σ 0 (A, F, y). Notice also that cork M (F ) = cork M |F ⊕M/F (F ). However by Lemma 4.6 we have that R 0 (F, y, L) Conversely, suppose A satisfies conditions (1) and (2). Let A be the matrix which has A as its rows, so what we have to prove is that π(A) = V . For any connected matroid M , we have that  We have Condition (2) of Theorem 4.12 says that any presentation has exactly one point in L 1 ∪ L 2 (the blue region in Figure 1) and exactly one point in L 3 ∪ L 4 (the red region in Figure 1), just as we said in Example 2.1. Condition (1) says that there is at most one point in L i for every i ∈ [4], and at most one point in L \ (L 1 ∪ L 2 ) and in L \ (L 3 ∪ L 4 ), but in this case this follows from condition (2). We end this section by using the previous theorem to understand how presentations behave under contractions.
Condition (2) of Theorem 4.12 applied to any of these matroids implies that there are exactly d − k points of A with ∞ in the F coordinates, because the cells of L corresponding to these cells extend to infinity in the e F direction. Let A F ⊆ A be the multiset of those points.
where ι F again means the inclusion L(V /F ) → L which sets the F coordinates to ∞. As the lattice of flats of M is isomorphic to the interval above F in lattice of flats of M , the conditions that Theorem 4.12 imposes on A F when applied to V are exactly the same as its conditions for presentations of V /F .

Matroid valuations
We will make use of the notion of matroid valuation, not to be confused with valuated matroids. This unfortunate similitude in names comes from the word "valuation" having pre-existing use in two different areas, respectively measure theory and algebra.
Given a polyhedron P ⊆ R n , let 1(P ) : R n → Z be its indicator function, defined by Definition 5.1. Let G be an abelian group, and f a function of a matroid taking values in G. We say that f is a (matroid) valuation if, whenever M 1 , . . . , M k are matroids and c 1 , . . . , c k integers such that For a general reference on matroid valuations, see [14]. We recount a few basic properties here. First, linear combinations of matroid valuations are again matroid valuations. Each nonempty intersection k∈K Q k is a matroid polytope, so discarding the terms with empty intersection gives a relation of form (5.1). Therefore such a subdivision of P M provides an "inclusion-exclusion" linear relation that a matroid valuation must satisfy. Repeating the same argument in the dual allows c (J),(r) (where the two indices are lists of length one) to be written as an alternating sum of terms s (I,J,K),(r−|J|+|I|,r,r) . We thus have where the sum is over choices of sets I i ⊆ X i and K i ⊇ X i for each i.
Submodularity implies that if rk(K) = rk(J) for some K ⊆ J, then also rk(K ∪ L) = rk(J ∪ L) for every L disjoint from K. Therefore, for any term of (5.2) in which K i ⊆ X i+1 for some i < k, with j ∈ X i+1 \ K i , inserting j into or removing it from K k gives another term which is equal with opposite sign. So we may cancel these terms, and by repeating the argument in the dual we may impose on the index set of the sum (5.2) the further conditions K i ⊆ X i+1 and I i ⊇ X i−1 . We have furthermore that any term with K i ⊆ I i+1 is zero, because if j ∈ K i \ I i+1 , submodularity is violated at X i ∪ {j} and X i+1 \ {j}.

Thus we can impose the condition K i ⊆ I i+1 on (5.2) as well. Under this condition all the sets in the indices form a single chain and we have
which is a valuation. It follows that c X,r (M ) is a valuation.
Recall the function τ defined in Equation (3.2).

Lemma 5.4. The function M → τ M (∅) is a matroid valuation.
Proof. By Philip Hall's theorem, the Möbius function µ(∅, F ) is a sum over the chains of cyclic flats from ∅ to F in CF, with a chain of length i weighted (−1) i . Therefore µ(∅, F ) cork(F ) can be written as a linear combination of the c X,r running over all chains of sets X = (X 0 = ∅, . . . , X k = F ) and all tuples r = (r 0 , . . . , r k ), the coefficient of c X,r being (d − r k )(−1) k . By Lemma 5.3, we conclude that M → τ M (∅) is a valuation.

The presentation space
The goal of this section is to describe the set of all presentations of a given valuated matroid V (Theorem 6.6). The techniques of the proof will give us further results such as Theorem 6.20, the converse of Proposition 3.6: if all facets of a regular subdivision correspond to transversal matroids, then the subdivision defines a transversal valuated matroid.   The support of the {0, ∞}-vectors within φ(U d,n ) give the set system presentations from Example 3.9.
The reason for calling Π(L) a presentation space is the following theorem. We prove the two directions of the equivalence in Theorem 6.6 separately. The easier one is Proposition 6.7, below. The other direction is Theorem 6.19. 6.2. Pseudopresentations. We recall the following characterization of transversal matroids in the form due to Ingleton [26]. Essentially the same characterization, but quantifying over all cyclic sets, was given earlier by Mason [32]. F 1 , . . . , F k the following inequality is satisfied:

Proposition 6.8. A matroid M is transversal if and only if for every collection of cyclic flats
Notice that for k = 2, this is the submodularity axiom of the rank function. We also remark that on substituting rk(J) = d − cork(J) in the above inequality, the d terms cancel out, and therefore a formally identical inequality is true where rk is replaced by cork and ≤ by ≥. To motivate this definition, note that it is a necessary condition for a presentation of M that the complements of its members be a pseudopresentation (see Proposition 3.7). The following lemma says that if a pseudopresentation fails to be the complements of a presentation, then the failure is "local", that is, there is a distinguished cyclic flat F such that the G i which extend F were poorly chosen. In other words, replacing every element in the pseudopresentation which does not extend F by its coclosure does not yield a presentation either. • cocl(G i ) = F for every i ∈ I • F F j for every j ∈ J.
• cork Proof. Suppose that such F does not exist but G 1 , . . . , G d are not the complements of a presentation. Then there is a set of indices Let k be the number of different elements of {cocl(G i ) : i ∈ I} and without loss of generality let that set be {F 1 , . . . , F k }. For j ∈ [k] let I j = {i ∈ I : cocl(G i ) = F j } and let m j = |I j |. The I j clearly partition I so we have that k j=1 m j = |I|.
x is a coloop of some G i , so in particular it is a coloop in K. Therefore we have that rk(K) =
Since the G i are pseudopresentation, we have that i∈I j G i consists of F j plus (possibly) some coloops. Since As we assume (F 1 , I 1 , ∅) is not a certificate as described in the lemma (as the tuple (F, I, J) in the statement), we have that Now for any 2 ≤ j ≤ k, let By inclusion-exclusion, we have that (The right hand side is counting the number of flats that contain F j and F j for some j < j.) Now notice that Similarly as before, we assume the conditions of the lemma are not satisfied for (F j , I j , J j ), so Adding all bounds for the m j and using Proposition 6.8 we get: which is a contradiction, as we assumed |I| > cork(K). If F = ∅ then H F is not a hyperplane, but in this event we will not use H F . We know rk V (F ) ≥ rk M (F ), so these cases are comprehensive.
Definition 6.14. An ascendent path is a finite or infinite sequence of such that for each i ≥ 0, either τ i is the last term of the sequence or τ i+1 is an ascendent step from τ i .
Let us give some intuition of what an ascendent path is. In each ascendent step, we go from the point x i in a colooples cell L M i and start going in a straight line within L in direction e F i until we change the cell of L where we are standing, so long as it is still is coloopless. If that change occurs immediately, that is, L M i+1 is of higher dimension than L M i and P M i+1 is a face of P M i , we keep going in the same direction (Case 1). If not, since L M i is bounded because M i is coloopless, then that change occurs at a face L M i+1 of L M i i.e. P M i is a face of P M i+1 (Case 2). This is the opposite of the last case in that In this case we may choose a new direction, however with the restriction above which is equivalent that We repeat this until the direction is ∅ (Case 0) or we leave the bounded region of L (Case 3). Again, what we will show for our later uses of this definitions, in Lemma 6.17, is that all ascendent paths terminate after finitely many steps (thus for example they cannot loop). The reason why we call the paths "ascendent" is Lemma 6.16.
be an ascendent path. Then, for Proof. We use descending induction on i. The base case is i = j, where F j \ F i is empty and therefore independent in any matroid.
If i < j then the ascendent step from (M i+1 , F i+1 , x i+1 ) belongs either to Case 1 or Case 2 of Definition 6.13. In Case 1,

which by induction hypothesis is independent in
which is what is needed.
In Case 2, first notice that By definition of ascendent step, F i+1 \ F i is independent in M i /F i = M i+1 \F i , so it is also independent in any restriction of M i+1 that contains it, in particular in M i+1 |(U ∪ F i+1 ). By the induction hypothesis, be an infinite ascendent path. Then the sequence of (x i ) i≥0 is nondecreasing when written in supportive coordinates with respect to x 0 , i.e. for every , the -th coordinate of x i is a nondecreasing function of i.
Proof. For each j ≥ 0 we have that x j+1 = x j + λ j e F j and thus (6.3)  F, x) such that for all τ ∈ S, an ascendent step from τ is also in S, then S is empty.
Proof. We argue that if (M, x, F ) is followed by an ascendent step of Case 1 in an ascendent path, then M can never appear subsequently in the path. This proves the result, because V has only finitely many initial matroids, and every step in Case 2 decreases the number of connected components of M so an infinite sequence of just Case 2 steps can't occur either.
By the assumption P M ⊆ H F of Case 1, there exists B ∈ B(M ) such that |B ∩ F | < rk M (F ). Suppose that (M, y, G) appears subsequently in the path. We have M = V y , so B ∈ B(V y ). By Lemma 6.16, rs x (y) consists of F plus possibly some other elements which by Lemma 6.15 are coloops of M | rs x (y). As B ∩ F is an independent set in M | rs x (y), we can extend it to a basisB ofM = M | rs x (y). Since F ⊆ rs x (y), we can arrange thatB contains rkM (F ) = rk M (F ) elements of F . Also, B contains all of the coloops ofM . Extend further to a basis B of M containingB. Since B contains fewer than rk M (F ) elements of F , this construction arranges that B ∩ rs x (y) is a strict superset of B ∩ rs x (y). By definition of relative support, this containment implies Since M = V x , we have that the basis A ∈ B(V ) causes V A − i∈A x i to take its minimum value exactly when A ∈ B(M ). In particular The final claim is clear.
We now use Lemma 6.17, either directly showing a contradiction or constructing an ascendent step (M , F , x ) from (M, F, x) that exhibits the same failure of presentation. Again, we break into the cases of Definition 6.13. Case 0. If F = ∅, contradiction is immediate because cork M (∅) = d. Case 1. For x = x+λe F with small enough λ, we have that rs x (A i ) = rs x (A i ) for any i ∈ I. Since for any set S that contains F we have that cork M (S) = cork M (S), we conclude that Then rs is not a presentation of M . So we can use Lemma 6.11 again to find F ∈ DF(M ) and I such that cocl M (rs x (A i )) = F where the conditions for presentation fail.
The only thing left to prove is that F \ F is independent in M/F , for (M , F , x ) to be indeed an ascendent step from (M, F, x). Notice that it follows from the proof of Lemma 6.11 that I ⊆ I. Then for any i ∈ I we have that F ⊆ rs x (A i ) ⊆ rs x (A i ). As cocl M (rs x (A i )) = F , then cocl M (F ) ⊆ F where it follows that F \ F is independent in M/F .

Case 3. Notice that
But this is a contradiction to the definition of φ M/F , which says that there is a presentation of M/F containing 6.5. Further consequences. A corollary of the above results is the converse of Proposition 3.6.

Theorem 6.20. A tropical linear space is in the Stiefel image if and only if all the facets in its dual subdivision are transversal.
Since the class of transversal matroids is closed under contractions of cyclic sets [8,Theorem 5.4] and arbitrary deletions, if V is transversal then so is any initial matroid V x which has no new coloops. Thus Theorem 6.20 can be sloganized: transversality is a local property of a tropical linear space. Proof. Let L be a tropical linear space dual to such a regular subdivision. By Theorem 6.20, L is in the Stiefel image so it has a presentation A. Consider the matrixÃ that replaces all finite entries of A by 0. Then π(Ã) is the Bergman fan of M , so M is transversal.  Figure 4 shows the presentation fan of each distinguished matroid: the fan from x 1 is the cone over the boundary of a square and the fan from x 2 is the cone over  the boundary of a triangle, while the fan from x 3 is the single point x 3 . So any matrix A ∈ π −1 (V ) must have one row in the red zone, another row in the blue zone and a third row lying exactly at the green point.

Strict gammoids and stable intersection
The first appearance of stable intersection of tropical varieties was as the fan displacement rule of Fulton and Sturmfels [22]. Speyer [45,Section 3] described the special case of stable intersection for tropical linear spaces in terms of Plücker coordinates.
d+d −n , provided that there exists some J for which the above formula yields (V ∩ stable V ) J < ∞.
In particular, for such a valuated matroid to exist we must have d + d ≥ n. By comparing this definition to Remark 3.13, we see that stable intersection is dual to stable sum, in the sense that general this containment can be strict (for example, whenever V = V ).
In matroid theory, the dual of a transversal matroid is commonly known as a strict gammoid. We allow a path to be zero edges long. The first sentence of Proposition 7.3 is due to Mason [31], the second to Ingleton and Piff [25].
Our work provides a valuated version of strict gammoids. We now describe these in terms of weighted directed graphs, akin to the graphs Speyer and Williams use to parametrize the tropical positive Grassmannian [44]. Consider a weighted directed graph Γ = ([n], E) with vertices [n] and where E is now a weight function E : [n] 2 → T which is 0 on the diagonal. The directed edges of the graph are exactly the pairs where E takes finite value. Let J ⊆ [n] be a subset of size d. Given a linking from a set B to J, the weight of that linking is the sum of the weights of all of the edges used in that linking. We call any such valuated matroid a valuated strict gammoid.
Proof. Consider A ∈ T (n−d)×n to be the matrix where the rows are indexed by I = [n] \ J and A i,j is the weight of the edge from i to j. In particular, A i,i is 0 for every i ∈ I. Let B ∈ [n] d and consider the tropical minor of A corresponding to the columns [n] \ B. A matching from those columns to the rows corresponds to picking edges such that every vertex in [n] \ B has exactly one edge coming in and all vertices in I have exactly one edge coming out. Taken together this is exactly a linking from B to J plus possibly some cycles in I \ B. The value of the term of that matching in the corresponding tropical minor is equal to the weight of the linking plus the weights of the cycles. However, as there are no negative cycles, removing the cycles (choosing the matching where for every vertex i in a cycle is matched with itself instead) the value of the corresponding term can only decrease. So the corresponding minor is equal to the minimum weight of a matching for B to J, that is, V B . This shows V is exactly the dual of π(A). Now if V is dual to a transversal valuated matroid π(A) with A ∈ T (n−d)×n , to construct the corresponding weighted graph Γ, let I be any basis of π(A) and let σ : [n − d] → I be a matching that achieves the minimum of π(A) I . Let Γ be the weighted directed graph where for every (i, j) ∈ I × [n] there is an edge from i to j with weight A σ −1 (i),j − A σ −1 (i),i . As σ achieves the minimum among matchings [n − d] → I there cannot be any negative cycles in Γ. So when the matrix A is constructed from Γ as described above, then A is obtained from A by subtracting A σ −1 (i),σ(i) from each entry of the row σ −1 (i). In particular π(A ) = π(A), so V is the valuated matroid associated to Γ.
As a corollary from Theorem 6.20 and Proposition 7.4 we get the following. (1) V is a valuated strict gammoid. Furthermore, Theorem 6.6 explicitly describes the space of all dtuples of tropical hyperplanes whose stable intersection is L(V ) and, through Proposition 7.4, all possible weighted directed graphs Γ representing V as a valuated strict gammoid. Notice that given a valuated strict gammoid V , collections of tropical hyperplanes whose stable intersection is L(V ) together with a matching σ are in bijection with weighted directed graph representations of V .

Other connections
8.1. Gammoids and maps. Among matroids, the class of gammoids is the minor-closure of either of the classes of valuated matroids or strict gammoids. So a class of valuated gammoids could be defined either as contractions of the transversal valuated matroids that are our main subject or as restrictions of the valuated strict gammoids of Section 7. Valuated gammoids are exactly the images of morphisms from free matroids in the sense of Frenk [21, §4.2], whose results are essentially a tropical formulation of earlier results from [29,33,35]. 8.2. Tropical convexity. As explained in Section 1, the tropical Stiefel map is one tropical counterpart of the map from a matrix to its rowspace. A different counterpart is the set of all T-linear combinations of a set of tropical vectors. This is known as the tropical cone. If the coefficients in the T-linear combination are further restricted to sum to 0 (the multiplicative identity element), we get the tropical convex hull. Tropical cones and convex hulls have been intensely studied from many points of view [1,3,11,15,23,27,42].
Tropical cones are usually not tropical linear spaces at all: [46, Theorem 16] describes when they are. However, tropical linear spaces are tropically convex [15,Theorem 7]. Lemma 4.1 implies the following. Thus, the tropical Stiefel map provides a bridge between these two objects, by giving a tropical linear space containing a given tropical cone (Corollary 8.1). If the tropical cone is r-dimensional and defined by r + 1 points, then the tropical Stiefel map provides an r-dimensional tropical linear space, which is smallest possible.
Every bounded cell of L(π(A)) is contained in the tropical cone T d ·A [20, Theorem 6.8]. More generally, T d · A contains the cells of L(π(A)) dual to coloop-free matroids, which is exactly the bounded part of L(π(A)) if V = U d,n .
8.3. Principal bundles. The Stiefel map was given the name "Stiefel" to reflect the fact that the space of tropical matrices maps to the space of valuated matroids just as the non-compact Stiefel manifold of d × n matrices of rank d maps to the Grassmannian of d-planes in n-space. Theorem 6.6 mirrors the classical fact that the non-compact Stiefel manifold is a principal GL d bundle over the Grassmannian, as we now explain. The only invertible matrices of tropical numbers are the generalized permutation matrices, those which have exactly one finite entry in every row and column, forming a group isomorphic to R S d . Theorem 6.6 implies that the space of d × n tropical matrices without too many infinities (Remark 3.2) has a deformation retract onto the Minkowski sum of the set of apices and the lineality space, which is a ramified R S d bundle over its image. The ramification arises because an apex can have equal rows.
It remains an open question to describe the topology of the image of the tropical Stiefel map. The above bundle perspective suggests a possible approach.