Tropical ideals do not realise all Bergman fans

Every tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.


Introduction
An ideal in a polynomial ring over a field with a non-Archimedean valuation gives rise to a tropical variety, either by taking all weight vectors whose initial ideals do not contain a monomial or, equivalently if the field and the value group are large enough [4,Theorem 4.2], by applying the coordinate-wise valuation to all points in the zero set of the ideal. In the middle of this construction sits a tropical ideal, obtained by applying the valuation to all polynomials in the ideal. This ideal is a purely tropical object, in that it does not know about the field or the valuation, and it contains more information than the tropical variety itself. For these reasons, tropical ideals, axiomatised in [6], were proposed as the correct algebraic structures on which to build a theory of tropical schemes. We review the relevant definitions below.
It was proved in [6] that tropical ideals, while not finitely generated as ideals-nor in any sense that we know of!-have a rational Hilbert series, satisfy the ascending chain condition, and define a tropical variety: a finite weighted polyhedral complex. Later in [7], it was shown that the top-dimensional parts of these varieties are always balanced polyhedral complexes. This leads to the following realisability question.
anced polyhedral complexes are realised by classical ideals has received much attention, especially in the case of curves (see e.g. [2,3,13]). But for general tropical ideals, very little is known about Question 1.1: for instance, no natural algebraic criterion that ensures that the variety is pure-dimensional is known. In fact, until recently we had no intuition as to whether tropical ideals are flexible enough that they can realise basically any balanced polyhedral complex, or rather more rigid, like algebraic varieties. In view of the following theorem, we now lean towards the latter intuition.
Theorem 5.2 Let M and N be loopless matroids of ranks a and b that do not have a quasi-product of rank a · b. Then, there exists no tropical ideal whose tropical variety is the Bergman fan of the direct sum of M and N , with all maximal cones having weight 1.
In particular, there exists no tropical ideal whose tropical variety is the Bergman fan of the direct sum of the Vámos matroid V 8 and the uniform matroid U 2,3 of rank two on three elements, with all maximal cones having weight 1.
In this theorem, a quasi-product of two loopless matroids is a matroid analogue of tensor products; see Sect. 4. The fact that the Vámos matroid V 8 and the uniform matroid U 2,3 have no quasi-product of rank 8 was proved by Las Vergnas in [5].
We believe that this theorem marks the beginning of an interesting research programme, which, in addition to the pureness and balancing questions mentioned above, asks which tropical ideals define matroids on the set of variables, and which matroids are, in this sense, tropically algebraic-See Problem 3.5 and Question 3.6.

Definitions and basic results on tropical ideals
Consider the tropical semifield (R := R∪{∞}, ⊕, • · ) with ⊕ := min and • · := +. Let R be a sub-semifield of R. The example most relevant to us is the Boolean semifield B := {0, ∞}, which is not only a sub-semifield but also a quotient of R.
and if, moreover, L satisfies the following elimination axiom: for i ∈ N and f, g ∈ L with f i = g i = ∞, there exists an h ∈ L with h i = ∞ and h j ≥ f j ⊕ g j for all j ∈ N , with equality whenever f j = g j .
The R-submodule L R of R N generated by L is a tropical linear space in R N and has the structure of a finite polyhedral complex; we denote its dimension as such by dim L.
If K is a field equipped with a non-Archimedean valuation onto R and if V ⊆ K N is a linear subspace, then the image of V under the coordinate-wise valuation is a tropical linear space in R N , but not all tropical linear spaces arise in this manner. Tropical linear spaces are well-studied objects in tropical geometry and matroid theory: the definition above is equivalent to that of [12], except that we allow some coordinates to be ∞. A tropical linear space L gives rise to a matroid M(L) in which the independent sets are those subsets A ⊆ N for which L ∩ (R A × {∞} N \A ) = {∞} N , and L is the set of vectors (R-linear combinations of valuated circuits) of a valuated matroid on M(L) [10]. With this setup, dim L = |N | − rk(M(L)). We will freely alternate between these different characterisations of tropical linear spaces.
. Tropical ideals were introduced by Maclagan and Rincón in [6] as a framework for developing algebraic foundations for tropical geometry. Tropical ideals are much better behaved than general ideals of the polynomial semiring R[x 1 , . . . , x n ], as we explain below. For a tropical ideal I, define its initial ideal relative to w as Note that in this paper we only consider weights w in R n , not in R n as in [6]. In other words, we do geometry only inside the tropical torus. Theorem 2.5 allows one to pass to monomial initial ideals and show that the Hilbert function H I (d) of a homogeneous tropical ideal I becomes a polynomial in d for sufficiently large d and also that homogeneous tropical ideals satisfy the ascending chain condition. Via homogenisation, one sees that both statements also hold for non-homogeneous tropical ideals (but, as in the classical setting, the theorem does not apply directly, since, for instance, when n = 1, in (1) (0 ⊕ x 1 ) = 0 generates an ideal-the entire semiring-with a smaller Hilbert function than any tropical ideal containing 0 ⊕ x 1 but not 0).
Furthermore, Maclagan and Rincón prove that tropical ideals have tropical varieties that are finite polyhedral complexes [6,Theorem 5.11].
is a bijection between V (I) and the intersection of V (I h ) with the zeroeth coordinate hyperplane, and we give V (I) the corresponding polyhedral complex structure.
The variety of a tropical ideal comes equipped with positive integral weights on its maximal polyhedra; this is inspired by [8,Lemma 3.4.7], and studied more in depth in [7].
. , x n ] be a tropical ideal, let σ be a maximal polyhedron of V (I), and let w be in the relative interior of σ . The multiplicity of σ in V (I) is defined as follows. First, let I ⊆ R[x ±1 1 , . . . , x ±1 n ] be the (tropical) ideal in the Laurent polynomial ring generated by I. After an automorphism of the Laurent polynomial ring given by x u → x Au with A ∈ GL n (Z), we can assume that the affine span of σ is a translate of span(e 1 , . . . , e d ) for some d. In this case, by [7, Lemma 6.2], the tropical ideal J : The multiplicity of σ is defined to be equal to this constant, called the degree of J .
Remark 2.8 A more coordinate-free version of Definition 2.7 is the following. Consider the linear span of σ , defined as n ] be the sub-semiring spanned by monomials x u of w-weight w · u equal to zero for all w ∈ span(σ ). Then, S itself is isomorphic to a Laurent polynomial semiring in n − d variables. The multiplicity of σ is the degree of the zero-dimensional tropical ideal in w (I ) ∩ S.
We will need the following results. We call I sat the saturation of I with respect to m := x 1 · · · x n , and we call I saturated with respect to m if I sat = I.
Proof That I sat is a tropical ideal containing I is straightforward from the definition. Since I sat ⊇ I, we have V (I sat ) ⊆ V (I). Conversely, let w ∈ V (I) and f ∈ I sat . Then, x u • · f ∈ I for some u ∈ N n , hence in w (x u • · f ) is not a monomial, and therefore, neither is in w f . This shows that V (I) = V (I sat ). That the multiplicities are the same follows from the fact that the multiplicities in V (I) are defined using I .
If is a polyhedral complex in R n and σ is a polyhedron in , the star star σ of at σ is a weighted polyhedral fan, whose cones are indexed by the cones τ of containing σ . The cone indexed by such τ is When I(I) is the collection of independent sets of a matroid M, we will say that I is a matroidal tropical ideal and that M is its associated algebraic matroid.
The independence complex of a tropical ideal I can be recovered from its variety V (I), at least if R = R.

2)
where π A : R n → R A is the coordinate projection onto the coordinates indexed by A. In particular, the independence complex I(I) depends only on the variety V (I).
Proof  In the classical setting, primality of an ideal implies matroidality. We do not know about a similarly appealing sufficient condition for matroidality of general tropical ideals.
x n ] is a prime ideal, where K is a field with a non-Archimedean valuation, then trop(J ) is a matroidal tropical ideal. Its associated algebraic matroid is the matroid that captures algebraic independence among the coordinate functions x 1 , . . . , x n in the field of fractions of K [x 1 , . . . , x n ]/J .

Problem 3.5 Find algebraic conditions on a tropical ideal that imply matroidality.
As shown in Example 3.4, any (classically) algebraic matroid is the algebraic matroid of a tropical ideal. However, in principle, it is possible that the class of matroids that are "tropically algebraic" is strictly larger than the usual class of algebraic matroids.

Quasi-products of matroids
To motivate the definition of quasi-products, let v 1 , . . . , v m be nonzero vectors in a vector space V and let w 1 , . . . , w n be nonzero vectors in a vector space W over the same field.
In the same manner, these define a matroid P with ground set [m] × [n]. One can check that P is in general not determined by M and N , i.e. the linear dependencies among the v i ⊗ w j cannot be read off from those among the v i and those among the w j . However, some features of P are predicted by M and N : for each fixed i ∈ [m], the linear dependencies among the vectors v i ⊗ w j , j ∈ [n] are precisely those recorded by N ; here we use that v i is nonzero. Similarly, for each j ∈ [n], the restriction of P to [m] × {j} is isomorphic to M. Furthermore, if B is a basis of M and C is a basis of N , then B × C is a basis of P. In particular, the rank of P is the product of the ranks of M and N . Following Las Vergnas, we use these observations to define quasi-products of general matroids, as follows. The properties of a quasi-product P of M and N imply that if B ⊆ [m] is a basis of M and C ⊆ [n] is a basis of N , then B × C is a spanning set of P, so the rank of P is at most the product of the ranks of M and N . By the discussion above, two matroids that are representable over the same field always admit a quasi-product whose rank is the product of their ranks. In general, however, a quasi-product with this property need not exist.

Not every Bergman fan is the variety of a tropical ideal
We now prove that not every balanced polyhedral complex can be obtained as the variety of a tropical ideal. Our counterexample will be the Bergman fan of a matroid; see [1] for details.  Bergman fans of matroids are the tropical linear spaces (more specifically, their part inside the torus R n ) that correspond to valuated matroids where the basis valuations all take values in B.
The following is our main result. Note that we do not require the polyhedral structure on V (I) coming from the Gröbner complex of the homogenisation of I to be equal to the fan structure on the Bergman fan described above.
To prove the theorem, in addition to the fundamental results from Sect. 2, we will need results relating V (I) to H I for any tropical ideal I. Proof The notion of stable intersection for tropical linear spaces was studied by Speyer in [12] when the underlying matroids of both tropical linear spaces were uniform matroids and later generalised by Mundinger [9] for arbitrary tropical linear spaces in R N . The stable intersection L ∩ st L is a tropical linear space contained in both L and L , and it has dimension a least dim L + dim L − |N | > 0, which implies the desired result. Proof Let I h ⊆ R[x 0 , . . . , x n ] be the homogenisation of I. Then, dim(I h ) d = dim I ≤d for all d ∈ N, and in particular dim(I h ) 1 = dim I ≤1 = n+1−H I (1) = n−r. Moreover, by applying Theorem 2.5 with a sufficiently general weight vector w, the Hilbert function of I h is also that of some monomial ideal J . We find that J contains precisely n−r of the n+1 variables x 0 , . . . , x n , and therefore all their multiples. This implies that dim J d ≥ n+d d − r+d d , where the last term counts monomials in the remaining r + 1 variables of degree d. We then have as desired.
The following proposition shows that the algebraic matroid of a Bergman fan B(M) (as in Proposition 3.2) is equal to the matroid M. We now present a key step towards proving our main result.  1 , . . . , x ±1 n ] generated by J . Take v to be a vector in the relative interior of τ . Since τ has multiplicity 1 in V (J ), in v (J ) ∩ S is zero-dimensional of degree 1 and contains no monomials. Hence, for any pair of distinct monomials x u , x u in S, in v (J ) ∩ S contains the binomial x u ⊕ x u . In particular, if {i = j} ⊆ F k \ F k−1 for some k then 0 ⊕ x −1 i x j ∈ in v (J ) ∩ S, and thus, x i ⊕ x j ∈ in v (J ). As J is homogeneous and saturated with respect to x 1 · · · x n , this implies that there is a polynomial of the form We conclude with the proof of the main theorem.
Proof of Theorem 5.2 Suppose that such an I exists, and denote O := M⊕N . We first argue that we may replace I by an ideal J that is homogeneous as well as saturated. To this end, let σ be a polyhedron in V (I) whose affine span is R·1 (which is contained in the lineality space The restriction Q|S 1 is spanned by all products of two elements in a basis of M(J 1 )|{x 1 , x 2 , . . . , x m }, and hence has rank at most a+1 2 . Similarly, the restriction Q|S 2 has rank at most b+1 2 . Hence, Q|S 3 has rank at least Since J is saturated, for each 1 ≤ i ≤ m, multiplication by x i yields an isomorphism between the matroid M(J 1 )|{y 1 , . . . , y n } ∼ = N and the restriction of Q to x i · {y 1 , . . . , y n } ⊆ S 3 . Similarly, for each 1 ≤ j ≤ n, the restriction of Q to y j · {x 1 , . . . , x m } is isomorphic to M. Hence, Q|S 3 is a quasi-product of M and N in the sense of Definition 4.1. But the assumption in the theorem is that such a quasi-product has rank strictly less than a · b, a contradiction. Hence, no such ideal I exists.
The second part of the main theorem is a direct consequence of the first part and Theorem 4.2 by Las Vergnas.

Concluding remarks
Using the result by Las Vergnas that U 2,3 and V 8 do not have a quasi-product of rank 8, we have showed that the Bergman fan of their direct sum is not the tropical variety of any tropical ideal, with weight 1 on all the maximal cones.
We do not know whether there exists a tropical ideal whose tropical variety is the Bergman fan of U 2,3 ⊕ V 8 as a set, without the condition that all weights be 1.
We also do not know whether B(V 8 ) itself is the tropical variety of any tropical ideal with weight one on the maximal cones. To study this question for a matroid M, one needs to develop the theory of symmetric squares of matroids, in a fashion similar to Las Vergnas's quasi-products from Sect. 4. But already for V 8 , this seems considerably harder than quasi-products of U 2,3 with V 8 .
Finally, we'd like to point out that for any m ≥ 3, the matroids U 2,m and V 8 do not admit a quasi-product of rank 8. Indeed, if P were such a quasi-product on [m] × [8], then for any basis C ⊆ [8] of V 8 the set [2] × C, which spans P, would have to be a basis. But then the restriction of P to [3] × [8] would be a quasi-product of U 2,3 and V 8 of rank 8, a contradiction to Las Vergnas's Theorem 4.2. This simple observation yields infinitely many matroids to which our Theorem 5.2 applies. However, it would be interesting to find more intricate families of pairs of matroids that do not admit quasi-products of the correct rank.