A "Staircase" formula for the Chern-Schwartz-MacPherson cycle of a matroid
Franquiz Caraballo Alba Jeffery Liu


Benchmarking the integration of hexagonal boron nitride crystals and thin films into graphene-based van der Waals heterostructures
    Bernd Beschoten
    September 9, 2024
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§ ABSTRACT 
    We provide a formula for the Poincaré dual of the Chern-Schwartz-MacPherson (CSM) cycle of a matroid in the Chow ring of the matroid. We derive the formula from the ℂ-realizable case and prove that it satisfies a contraction-deletion formula. From this fact, we prove it holds for all matroids, confirming a conjecture of Fife and Rincón.




§ INTRODUCTION


The Chern-Schwartz-MacPherson (CSM) class generalizes the Chern class of the tangent bundle of a variety. In particular, it is defined even when the variety is singular or non-proper. One interpretation of the CSM class of a variety is in terms of a natural transformation between the functor of constructible functions on varieties and the Chow group functor, for more details, see <cit.>. In the case where V is the complement of a hyperplane arrangement 𝒜 in ^d, a combinatorial formula for c_SM(V) in terms of the characteristic polynomial of the underlying matroid was given separately by Aluffi in <cit.> and Huh in <cit.>.

De Concini and Procesi defined the notion of a wonderful model of a hyperplane arrangement with respect to a building set in <cit.>. In this paper, we will use the term wonderful model of 𝒜 to refer to the wonderful model with respect to the maximal building set. Later, Feichtner and Yuzhvinsky <cit.> define the Chow ring of a matroid following the ideas of <cit.> and the correspondence between the Chow ring of a nonsingular proper toric variety X and Minkowski weights on its associated fan Σ proven in <cit.>. In <cit.>, Adiprasito, Huh and Katz bring these concepts together and prove that A^*(M) and MW_*(B(M)) are dual to each other, in analogy to Poincaré duality for the toric case and the realizable case. This gave the necessary setting for López de Medrano, Rincón, and Shaw to define an analogous CSM cycle for matroids.

In <cit.>, the authors define (M) for a matroid M and prove in Theorem 3.1 that when M is realizable over ℂ, its CSM cycle corresponds to the geometric CSM class c_SM(𝒲∖𝒜) of the complement of the arrangement in the wonderful model under the identification of MW_*(B(M)) ≅ A_*(𝒲). However, in the geometric setting c_SM(𝒲∖𝒜) should be Poincaré dual to some element c^SM(𝒲∖𝒜) ∈ A^*(𝒲), since 𝒲 is proper and nonsingular. In the combinatorial setting, the Minkowski weights on the Bergman fan of M and the Chow ring of M satisfy Poincaré duality, so there should be a class c^SM(M) ∈ A^*(M) with c^SM(M) ∩ 1_M = (M). In Section 3 we prove


    The CSM class of the complement of the arrangement 𝒜 in its wonderful model 𝒲 is given in terms of the Chow Ring generators by the formula 
    
    c_SM(𝒲∖𝒜)    = ∏_r=1^(E)(1 - ∑_F ∈ℒ
    (F) ≥ r x_F) ∩ [𝒲],

    where M is the matroid associated to 𝒜.


In <cit.>, Fife and Rincón conjectured that the degree k part of c^SM(M) is given by the sum 

    ∑_|ℱ|=k C_ℱ x_F_r_1… x_F_r_k

indexed over all size k multisubset chains ℱ = {F_r_1⊆…⊆ F_r_k} of flats of M, with coefficients C_ℱ on each monomial 

    C_ℱ := (-1)^k ∏_i = 1^k (r_i - i + 1)∏_r = 1^(E) m_ℱ(r)! = (-1)^k (r_1) (r_2 - 1) ⋯ (r_k - k + 1)m_ℱ(1)! m_ℱ(2)! ⋯ m_ℱ((E))!.

where r_i = (F_r_i) and m_ℱ(r) counts the multiplicity of r among {r_1, …, r_k}.

Theorem <ref> proves this conjecture in the case of matroids realizable over ℂ and Corollary <ref> generalizes the answer to arbitrary matroids.

In Section 2 we recall the required combinatorial and algebro-geometric background for the rest of the paper. In Section 3 we prove that our formula does indeed compute c^SM(𝒲∖𝒜) in the case where M is realized by the hyperplane arrangement 𝒜. In Section 4 we define 

    (M) := ∏_r=1^(E)(1 - ∑_F ∈ℒ
    (F) ≥ r x_F) ∈ A^*(M),

the staircase class of a matroid, which generalizes the formula provided in Section 3 to the case where M is an arbitrary matroid and prove in Theorem <ref> that this class satisfies a contraction-deletion formula similar to that of (M). The name comes from the visual structure of the class when written out explicitly; in the case where M = U_3,3 with underlying set E = {0,1,2,3},

    (M) =     (1     - x_0 - x_1 - x_2     - x_01 - x_02 - x_12     - x_012) 

    ·    (1          - x_01 - x_02 - x_12     - x_012) 

    ·    (1               - x_012)



Finally, we prove

    For an arbitrary matroid M, (M) ∩ 1_M = (M).

The proof of this statement uses the contraction-deletion formula, the interpretation of the Chow ring in terms of piecewise polynomials <cit.> and induction on the number of elements of M, which completes the proof of Conjecture 5.2.4 in <cit.>.



§ BACKGROUND


 
    A matroid M is a finite set E, together with a non-negative interger valued rank function on its subsets : 2^E →^≥ 0, satisfying the properties:
    
        
  * for all subsets A ⊆ E, (A) ≤ |A|.
        
  * for all A,B ⊆ E, if A⊆ B, then (A) ≤(B).
        
  * for all A,B ⊆ E, (A∪ B) + (A ∩ B) ≤(A) + (B)
    
    A subset F ⊆ E is called a flat of the matroid if for all i ∈ E ∖ F, (F) < (F ∪{i}), or equivalently if it is maximal with respect to inclusion among other subsets of the same rank (F).
    The set of all flats forms a lattice under inclusion. The set of all non-empty flats will be denoted by ℒ(M), and the set of all proper, non-empty flats will be denoted by ℒ̂(M). We will also write just ℒ and ℒ̂ to save space if the matroid is understood.



    Given a matroid M=(E,),
    
        
  * an element i ∈ E is a loop if (i) = 0
        
  * an element i ∈ E is a coloop if every minimal set with rank (E) contains i. In other words, i is a coloop if every basis of M contains i.
        
  * A matroid M=(E,) is simple if its rank 1 flats are precisely the singletons {i}, for all i ∈ E.
    



    Given an element i ∈ E and a matroid M = (E, ) on E, we define the deletion M ∖ i as the matroid (E ∖{i},|_2^E ∖{i}) and the contraction M / i as the matroid (E ∖{i}, ') with '(A) = (A ∪{i}) - ({i}).



    Given a collection of hyperplanes 𝒜 = {H_0, …, H_n} in ^d_ (or hyperplanes through the origin in ^d+1), the function : 2^{0, …, n}→^≥ 0 defined by (S) = codim( ⋂_i ∈ S H_i) is a matroid on the ground set {0, …, n}.

    Given a set of vectors {v_0, …, v_n} in ^d+1, the rank function (S) = (span{v_i | i ∈ S}) defines a matroid. This yields the same matroid as the previous example by taking the normal hyperplane to each vector.

    Matroids arising in this way from a hyperplane arrangement in ^d_ (or a vector arrangement in ^d+1) are called -realizable. Note that there exists matroids which are not realizable.


In the realizable case, if M = ({0,…,k},) is represented by the set of vectors {v_0, …, v_k} in ℂ^n, then M  i is represented by the set of vectors {v_0, …, v_k}∖{v_i} in ℂ^n and M / i is represented by {v_0 + ⟨v_i|,⟩…, v_k + ⟨v_i|}⟩∖{⟨v_i|}⟩ in ℂ^n / ⟨v_i|$⟩.
    

    Given a finite set E, the matroid M = (E,|_|) is realizable over ℂ by 𝒜 = {V(x_i) | i ∈ E} in (ℂ^E).

    Consider the arrangement of four lines in ^2:
    
    H_0    = V(x_0)
    
            H_1    = V(x_1)
    
            H_2    = V(x_1 - x_2)
    
            H_3    = V(x_2)

    This arrangement is depicted in Figure <ref> as viewed from the chart x_0 + x_1 + x_2 ≠ 0.
    The rank function of the corresponding realizable matroid has values
    
    (∅)   =0
    (0)=(1)=(2)=(3)   =1
    (01)=(02)=(03)=(12)=(13)=(23)=(123)   =2
    (012)=(013)=(023)=(0123)   =3

    and its flats has the lattice structure
    

    0123 
    
    	01     02     03     123 
    
    	0     1     2     3 
       ∅[from=2-1, to=1-2]
    	[from=2-2, to=1-2]
    	[from=2-3, to=1-2]
    	[from=2-4, to=1-2]
    	[from=3-1, to=2-1]
    	[from=3-1, to=2-2]
    	[from=3-1, to=2-3]
    	[from=3-2, to=2-1]
    	[from=3-2, to=2-4]
    	[from=3-3, to=2-2]
    	[from=3-3, to=2-4]
    	[from=3-4, to=2-3]
    	[from=3-4, to=2-4]
    	[from=4-2, to=3-1]
    	[from=4-2, to=3-2]
    	[from=4-2, to=3-3]
    	[from=4-2, to=3-4]


For an in depth look at matroids, we reference <cit.>.


    Let 𝒜 = {H_0, …, H_n} be hyperplane arrangement in ^d. For every flat F in the matroid associated to 𝒜, there is a linear subvariety corresponding to the intersection L_F = ∩_i ∈ F H_i. 
    For an arbitrary arrangement 𝒜, its De Concini-Procesi wonderful model 𝒲 is a variety constructed from ^d by a sequence of blow-ups at each the intersections by order of dimension: 
    
  *  First blow up the points {L_F| F ∈ℒ̂, (F) = d}.
        
  *  Then blow up the proper transform of the lines {L_F| F ∈ℒ̂, (F) = d-1}.
        
  *  Continue in this manner, at each step i = 0, …, d-1, blowing up the proper transform of the i dimensional linear subspaces {L_F| F ∈ℒ̂, (F) = d-i}.
    
    
    Let π: 𝒲→^d denote this blow-up map. π is an isomorphism on the complement of the arrangement C(𝒜). On the 𝒲 there is a divisor corresponding to each flat of the matroid. All collections of these divisors intersect transversely if their flats are comparable, and are disjoint otherwise. See <cit.> for details.

    Given a matroid M on a ground set E, we associate its Chow Ring 
    
    A^*(M) := [x_F | F ∈ℒ̂(M)](I_M + J_M),

    where I_M and J_M are the ideals
    
    I_M    := ({x_Fx_F'|if F and F' are not comparable})
    
            J_M    := ( {∑_F ∈ℒ̂
     F ∋ i x_F - ∑_F ∈ℒ̂
     F ∋ j x_F| i,j ∈ E}).

    An equivalent presentation over all non-empty flats is 
    
    A^*(M) := [x_F | F ∈ℒ(M)](I_M + J_M)

    where I_M and J_M are the ideals
    
    I_M    := ({x_Fx_F'|if F and F' are not comparable})
    
            J_M    := ( {∑_F ∈ℒ
     F ∋ i x_F| i ∈ E})

    note that the extra generator x_E equals x_E = -∑_F ∈ℒ̂
 F ∋ i x_F for all i ∈ E.

When a matroid is-realizable, arising from a hyperplane arrangement𝒜, its Chow Ring is isomorphic to the Chow Ring of the wonderful modelA^*(𝒲)in the geometric sense, with eachx_Fcorresponding to the class of the divisor associated to the proper flatF, andx_E = -π^*(h), minus the pullback of the hyperplane class in^d.


    Let M be the matroid in Example <ref>
    Its Chow Ring is presented by the quotient ring
    
    [x_0, x_1, x_2, x_3, x_01, x_02,x_03,x_123,x_0123](I_M + J_M)

    with relations given by the ideals
    
    I_M = (   x_0x_1, x_0x_2, x_0x_3, x_0x_123, x_1x_2, x_1x_3, x_1x_02, x_1x_03, x_2x_3, x_2x_01, x_2x_03,
            x_3x_01, x_3x_02,
       x_01x_02, x_01x_03, x_01x_123,
            x_02x_03, x_02x_123,
            x_03x_123)
    
            J_M = (   x_0 + x_01 + x_02 + x_03 + x_0123, x_1 + x_01 + x_123 + x_0123,
       x_2 + x_02 + x_123 + x_0123, x_3 + x_03 + x_123 + x_0123)

    Let E be a finite set and for S ⊂ E define e_S be the element of ℝ^E given by e_E(i) = 1 for all i ∈ S. Given a matroid M = (E, ), we define B̂(M), the affine Bergman fan of M as the fan in ℝ^E whose cones are of the form
    
    σ̂_ℱ := {∑_i = 1^k a_ie_F_i| a_i ≥ 0 },

    where ℱ = {∅⊊ F_1 ⊊…⊊ F_k ⊂ E} is a flag of non-empty flats of M. We define the image of this fan under the projection ℝ^E →ℝ^E / ℝ e_E to be B(M), the Bergman fan of M. Note that this projection maps σ̂_ℱ and σ̂_ℱ∪{E} to the same cone, where ℱ is a flag of proper non-empty flats. We denote the cone corresponding to ℱ in B(M) by σ_ℱ.


The following definitions of Chow ring and Minkowski weights of a fanΣare explored in detail in Chapter5of <cit.>. We reproduce the basic definitions and facts for the reader:


    Given a unimodular polyhedral fan Σ in a latticed vector space N_ℝ = N ⊗_ℤℝ we define the Chow ring of Σ to be
    
    A^*(Σ) :=    ℤ[{x_ρ | ρ∈Σ(1)}]/(I_Σ + J_Σ),

    where I_Σ consists of monomials x_ρ x_ρ' with ρ, ρ' ∈Σ(1) such that there is no cone σ in Σ such that ρ and ρ' are both codimension 1 subcones of σ and J_Σ consists of linear forms
    
    ∑_ρ∈Σ(1)⟨m,v_ρ|x⟩_ρ,

    where m ∈_ℤ(N, ℤ) and v_ρ is the generator of ρ in N.

    In the specific case where Σ is the (affine) Bergman fan of a matroid M,
    
    A^*(B̂(M)) ≅ A^*(B(M)) ≅ A^*(M)

    by the isomorphisms given by x_σ̂_F↦ x_σ_F↦ x_F, for F ≠ E and x_σ̂_E↦ - ∑_F ∋ i x_σ_F.

    A weighted fan in N_ℝ is a polyhedral fan Σ, of pure dimension k, and a weight function ω:Σ(k) →ℤ. We say that (Σ, ω) is balanced if for all τ∈Σ(k-1),
    
    ∑_σ > τω(σ) v_σ / τ∈⟨τ|,⟩

    where v_σ / τ is the generator of σ + ⟨τ|$⟩ inN_ℝ / ⟨τ|$⟩
    Given a fan Σ of pure dimension n, we define k-dimensional Minkowski weights on Σ to be the set of balanced weighted subfans of Σ.

    Given a pure n-dimensional, balanced, unimodular fan Σ in N_ℝ, A^k(Σ) ≅ MW_n - k(Σ). We denote the isomorphism by _∩ 1_Σ, where 1_Σ is the function σ↦ 1 for all σ∈Σ(n). In the case where Σ = B(M) for some matroid M, we simply write 1_M for 1_B(M). In particular, Remark <ref> and the statement above imply that for all matroids M, MW_*(B̂(M)) ≅ MW_*(B(M)).

    In the case where Σ is the fan associated to a proper, nonsingular toric variety X(Σ), A^*(Σ) ≅ A^*(X(Σ)) and A_*(X(Σ)) ≅ MW_*(Σ), see <cit.> for details.


§ A “STAIRCASE" CSM CLASS FORMULA FOR REALIZABLE MATROIDS

The Chern-Schwartz-MacPherson (CSM) class generalizes the Chern class of the tangent bundle of a variety. More explicitly, letVar_ℂbe the category of varieties with proper maps as morphisms andAbbe the category of abelian groups.


    There exists a unique natural transformation c_*:𝒞_* → A_* such that for a proper, nonsingular variety V, c_*(1_V) = c(TV) ∩ [V] where 𝒞_*: Var_ℂ→ Ab is the functor of constructible functions and A_*: Var_ℂ→ Ab is the Chow group functor. See Theorem 1 in <cit.> for details.

    For a (possibly singular and/or non-proper) subvariety X embedded in a nonsingular proper variety V, define c_SM(X) = c_*(1_X) ∈ A_*(V).


López de Medrano, Rincón, and Shaw defined an analogous CSM cycle for matroids, and proved that when a matroid is realizable overℂ, its CSM cycle corresponds to the geometric CSM classc_SM(𝒲 ∖𝒜)of the complement of the arrangement in the wonderful model under the identification ofA^*(M) ≅A^*(𝒲)(Theorem 3.1 in <cit.>).

A formula for the CSM cycle of a matroid in terms of the generators of the Chow Ring was conjectured by Fife, T. and Rincón, F in private communication, according to Conjecture 5.2.4 in <cit.>.

In this section, we prove such a formula for the CSM class of the complement of the arrangement by geometric means, thus resolving the conjecture for all realizable matroids.

In the next section, we extend this formula to all matroids, and prove that it gives the same CSM cycle as defined by López de Medrano, Rincón, and Shaw.

To proceed, we utilize some geometric formulas for the CSM class due to Aluffi:
 (Lemma 1.3 in <cit.>)
    Let X ⊆ Y be a nonsingular varieties, and π: Ỹ→ Y the blow-up of Y along X, and X̃ the exceptional divisor. If X is a complete intersection of hypersurfaces Z_1, …, Z_d meeting transversally in Y, then the Chern class of the tangent bundle of the blow up Ỹ is related to c(T_Y) by the formula
    
    c(T_Ỹ) = (1 + X̃)(1 + π^*Z_1 - X̃)⋯(1 + π^*Z_d - X̃)(1 + π^*Z_1)⋯(1 + π^*Z_d)π^*c(T_Y)
 (Theorem 1 in <cit.>)
    Let X ⊂ Y be a subvariety of a nonsingular variety, with X = D_1 ∪…∪ D_n a normal crossings divisor with smooth components. The CSM class of the complement Y∖ X is
    
    c_SM(Y∖ X)    = c(Ω^1_Y(log X)^∨ ) ∩ [Y] 
       = c(T_Y)(1+D_1) ⋯ (1+D_n)∩ [Y]

    The CSM class of the complement of the arrangement in its wonderful model is given in terms of the Chow Ring generators by the formula 
    
    c_SM(𝒲∖𝒜)    = ∏_r=1^(E)(1 - ∑_F ∈ℒ
    (F) ≥ r x_F) ∩ [𝒲]


The wonderful model 𝒲 is constructed by a sequence of blow-ups of ^d at the intersection of hyperplanes determined by each flat F ∈ℒ̂. If we blow-up according to order by rank, highest to lowest (i.e. by dimension, lowest to highest), the locus of blow-up is intersection (F) many hyperplanes at the intersection determined by F minus the exceptional divisors already blown up in the intersection (corresponding to the flats F' which contain F. That is, when all blow-ups are complete, each of the (F) hyperplanes in locus of blow-up pulls back to π^* h -∑_F' ∈ℒ̂
 F' ⊋ F x_F' in A^*𝒲, where h is the class of the hyperplane in ^d. By applying the blow-up formula (Lemma <ref>) according to each flat F ∈ℒ̂,

    c(T_𝒲) = ∏_F ∈ℒ̂[(1 + x_F) (1 + (π^* h -∑_F' ∈ℒ̂
     F' ⊋ F x_F') - x_F)^(F)(1 + (π^* h -∑_F' ∈ℒ̂
     F' ⊋ F x_F'))^(F)]π^*c(T_^d).


The complement of the arrangement in 𝒲 is the complement of the union of all divisors x_F for all F ∈ℒ̂, which is normal crossings with smooth components by construction. By the complement formula (Lemma <ref>), we get

    c_SM(𝒲∖𝒜)    = c(T_𝒲)∏_F ∈ℒ̂ (1 + x_F)∩ [𝒲]
       = 1∏_F ∈ℒ̂ (1 + x_F)∏_F ∈ℒ̂[(1 + x_F) (1 + (π^* h -∑_F' ∈ℒ̂
     F' ⊋ F x_F') - x_F)^(F)(1 + (π^* h -∑_F' ∈ℒ̂
     F' ⊋ F x_F'))^(F)]π^* c(T_^d) ∩ [𝒲].

By cancellation, and π^*c(T_^d) = (1 + π^* h)^d+1 = (1 + π^* h)^(E), we get

    c_SM(𝒲∖𝒜) = …   = ∏_F ∈ℒ̂[(1 + (π^* h -∑_F' ∈ℒ̂
     F' ⊋ F x_F') - x_F)^(F)(1 + (π^* h -∑_F' ∈ℒ̂
     F' ⊋ F x_F'))^(F)]π^*c(T_^d) ∩ [𝒲]
       = ∏_F ∈ℒ̂[(1 + (π^* h -∑_F' ∈ℒ̂
     F' ⊋ F x_F') - x_F)^(F)(1 + (π^* h -∑_F' ∈ℒ̂
     F' ⊋ F x_F'))^(F)](1 + π^* h)^(E)∩ [𝒲].

We can substitute x_E := -π^*h to simplify notation:

    c_SM(𝒲∖𝒜) = …   = ∏_F ∈ℒ̂[(1 + (-x_E -∑_F' ∈ℒ̂
     F' ⊋ F x_F') - x_F)^(F)(1 + (-x_E -∑_F' ∈ℒ̂
     F' ⊋ F x_F'))^(F)](1 -x_E)^(E)∩ [𝒲]
       = ∏_F ∈ℒ̂[(1 - ∑_F' ∈ℒ
     F' ⊇ F x_F')^(F)(1 -∑_F' ∈ℒ
     F' ⊋ F x_F'))^(F)](1 -x_E)^(E)(1)^(E)∩ [𝒲]
       = ∏_F ∈ℒ(1 - ∑_F' ∈ℒ
     F' ⊇ F x_F'1- ∑_F' ∈ℒ
     F' ⊋ F x_F')^(F)∩ [𝒲].

Next, we compute the coefficients in the formal expansion of the above expression in [x_F | F ∈ℒ]. The coefficient of each monomial ∏_F ∈ℱ x_F (where ℱ some multisubset of ℒ) may be extracted as follows:

ℱ must be a chain, since otherwise  ∏_F ∈ℱ x_F = 0 has empty intersection in A^*𝒲. Choose a maximal flag {∅ = F_0 ⊊ F_1 ⊊…⊊ F_d ⊊ F_d+1 = E} (where (F_i) = i) containing the chain ℱ as a multisubset. We may rewrite the monomial as

∏_F ∈ℱ x_F = x_F_r_1⋯ x_F_r_k.

where k = |ℱ| is the degree of the monomial (also, the size of the multiset), and r_1 ≤…≤ r_k is a monotonic sequence. In other words, we re-index the multiset according to the order in the flag by inclusion: 
ℱ = {F_r_1, …, F_r_k},    F_r_1⊆…⊆ F_r_k.

Let m_ℱ(r) = |{F ∈ℱ|(F) = r}| count the multiplicity of F_r in the multisubset ℱ, i.e. the number of occurrences of r in {r_1, …, r_k}.

For all F not included in the flag, we may set x_F = 0. This has no effect on ∏_F ∈ℱ x_F, and only kills some other monomials). If F is not in the flag, the fraction (1 - ∑_F' ⊇ F x_F'1- ∑_F' ⊋ F x_F')^(F) becomes 1. Otherwise, when F = F_r in the flag, it becomes (1 - ∑_i = r^(E) x_F_i1- ∑_i = r+1^(E) x_F_i)^r. The expression then simplifies by cancellation (we indicate the result of killing the non-flag divisors by ↦):

    ∏_F ∈ℒ(1 - ∑_F' ⊇ F x_F'1- ∑_F' ⊋ F x_F')^(F)   ↦∏_r = 1^(E)(1 - ∑_i = r^(E) x_F_i1- ∑_i = r+1^(E) x_F_i)^r       = ∏_r = 1^(E)(1 - (x_F_r + x_F_r+1 + … + x_E)1- (x_F_r+1 + … + x_E))^r
       = ∏_r = 1^(E)(1 - ∑_i = r^(E) x_F_i)       = ∏_r = 1^(E) (1 - (x_F_r + … + x_E)).

The coefficient of the monomial x_F_r_1⋯ x_F_r_k may be deduced by counting how many times it the appears in the expansion.[alternatively, we can use the Taylor's Theorem to compute the coefficient by taking the appropriate partial derivatives at zero.] The count amounts to choosing from which factor each x_F_r_i comes, then dividing to account for identical permutations. Note that each -x_F_r_i appears as a term only in the first r_i factors. There are r_1 possible choices for x_F_r_1, then r_2 - 1 many remaining choices for x_F_r_2, and so on (i.e. r_i - i + 1 choices for x_F_r_i once the previous i-1 variables have been chosen). Therefore, the coefficient on x_F_r_1⋯ x_F_r_k is

    C_ℱ := (-1)^k ∏_i = 1^k (r_i - i + 1)∏_r = 1^(E) m_ℱ(r)! = (-1)^k (r_1) (r_2 - 1) ⋯ (r_k - k + 1)m_ℱ(1)! m_ℱ(2)! ⋯ m_ℱ((E))!.

Denote this coefficent by C_ℱ for each multisubset ℱ. These are precisely the coefficients conjectured by Fife and Rincón.
Furthermore, an alternate expression

    ∏_r=1^(E)(1 - ∑_F ∈ℒ
    (F) ≥ r x_F)

yields the same coefficients on every monomial, since it has the same form after killing non-flag divisors: ∏_r=1^(E)(1 - ∑_(F) ≥ r x_F) ↦∏_r = 1^(E)(1 - ∑_i = r^(E) x_F_i). By comparing coefficients, we conclude that it is equal to the original CSM class:

    c_SM(𝒲∖𝒜) = …   = ∏_F ∈ℒ(1 - ∑_F ∈ℒ
     F' ⊇ F x_F'1- ∑_F ∈ℒ
    F' ⊋ F x_F')^(F)∩ [𝒲]
       = ∑_ℱ multisubset of ℒ C_ℱ∏_F ∈ℱ x_F ∩ [𝒲]
       = ∏_r=1^(E)(1 - ∑_F ∈ℒ
    (F) ≥ r x_F) ∩ [𝒲].

    The formula conjectured by Fife and Rincón holds for realizable matroids.

     Let 𝒜 be the arrangement in Example <ref>. The CSM class of the complement of the arrangement in its wonderful compactification by the previous result is
    
    c_SM(𝒲∖𝒜)=    (1-x_0-x_1-x_2-x_3-x_01-x_02-x_03-x_123-x_0123)
    ·   (1-x_01-x_02-x_03-x_123-x_0123)
             ·(1-x_0123) ∩ [𝒲]
    
            =    (1-x_123-x_0123) ∩ [𝒲].


§ RELATION TO THE CSM CYCLE OF A MATROID


The result from the previous section motivates the following definition for all matroids, not necessarily realizable.

    Given a loopless matroid M, define the invariant (M) in its Chow Ring A^*(M) by the formula
    
    (M) := ∏_r=1^(E)(1 - ∑_(F) ≥ r x_F) ∈ A^*(M).

    In the case where M has a loop, define (M) = 0.
    
We prove in this section that(M) ∩1_Mis the CSM cycle of a matroid defined by López de Medrano, Rincón, and Shaw. 
Note that the coefficientsC_ℱconjectured by Fife and Rincón are precisely the coefficients in the expansion of(M), so this is equivalent to the statement of their conjecture in full generality.

First we show that this invariant respects certain deletion and contraction operations on a matroid.


    Let M be a matroid on ground set E with rank function _M. For any i ∈ E, we can construct the matroid M∖ i, called the deletion of M by the element i. This is the matroid on the set E∖ i with rank function _M∖ i: 2^E∖ i
    _M∖ i(S) := _M(S)

    For any i ∈ E, we can construct the matroid M / i, called the contraction of M by the element i.
    This is the matroid on the set E∖ i with rank function _M∖ i: 2^E∖ i
    _M / i(S) := _M(S ∪ i) - _M(i)


 §.§ Contraction-deletion of 


For a given elementi ∈Ewhich is not a coloop, we obtain a projectionδ: B(M) →B(M ∖i). By Proposition 2.22 in <cit.>,δis a tropical modification along some rational functionϕ; therefore,δinduces a pullbackδ^*:Z_*(B(M ∖i) →Z_*(B(M))and sincediv_B ∖i(ϕ) = B(M / i), we also obtain a pullbackZ_*(B(M / i)) →Z_*(B(M ∖i)). These have corresponding ring/group homomorphisms in the context of the Chow rings, see Section 4.2 for details.


    To distinguish the divisors of A^*(M), A^*(M ∖ i), and A^*(M/i), we label them by
    
    A^*(M)    = ℤ[x_F | F ∈ℒ(M)]/(I_M + J_M)
    
            A^*(M∖ i)    = ℤ[y_F | F ∈ℒ(M∖ i)]/(I_M ∖ i + J_M ∖ i)
    
            A^*(M/ i)    = ℤ[z_F | F ∈ℒ(M/i)]/(I_M/i + J_M/i)

    
    i.e. with variable name x, y, z respectively. Define the deletion pullback (a ring homomorphism) δ̅^*: A^*(M ∖ i) → A^*(M) on each generator y_F by 
    
    δ̅^*(y_F) = ∑_F' ∈ℒ(M) 
     F'∖ i = F x_F'.


    Define the element y_i ∈ A^*(M ∖ i) by
    
    y_i := - ∑_F ∈ℒ(M∖ i) 
     F ∉ℒ(M) 
     F ∪i∈ℒ(M) y_F.


    Multiplication by y_i defines a map A^*(M / i) → A^*(M ∖ i)
    z ↦ y_i ·ι(z)

    where ι(z_F) = y_F (which replaces the variables z with y). This is an embedding A^*(M / i) as the subgroup y_i · A^*(M ∖ i):

    By composition we also obtain a map δ̅^* (y_i ·ι(_)): A^*(M / i) → A^*(M).

    The invariant (M) satisfies the contraction-deletion formula:
    
    (M) = δ̅^*((M∖ i)) - δ̅^* (y_i ·ι((M/i)))

We partition the nonempty flats of M according to the flowchart:


    is i ∈ F?
    is F∪{i} a flat?      is F-{i} a flat (or ∅)?
    𝒯   𝒮   𝒰   𝒱["no"', from=1-1, to=2-1]
    	["yes", from=1-1, to=2-3]
    	["no", from=2-1, to=3-1]
    	["yes", from=2-1, to=3-2]
    	["yes", from=2-3, to=3-3]
    	["no", from=2-3, to=3-4]



The sets 𝒯, 𝒮, 𝒰, 𝒱 are disjoint and partition ℒ(M). 

Denote also 𝒰∖ i := {F∖ i | F ∈𝒰} and 
𝒱∖ i := {F∖ i | F ∈𝒱}.

Note the following reasons for this partition:

  * 𝒰∪𝒱 consists of all flats containing the deleted element i. Among these, 𝒰 are the flats that decrease in rank when i is deleted. On the other hand, 𝒱 are the flats that remain the same rank when i is deleted.

    
  * S ∪𝒱∖ i = ℒ(M/i) are the flats of the contraction M/i.
    
    
  * 𝒯∪𝒮∪𝒱∖ i = ℒ(M ∖ i) are the flats of the deletion M∖ i.
    
    
  * 𝒰∖ i = 𝒮∪{∅}.

This partition determines the behavior of the divisors under the map δ̅^*: 

  *  If F ∈𝒯, then δ̅^*(y_F) = x_F. 
    
  *  If F ∈𝒮, then δ̅^*(y_F) = x_F + x_F ∪{i}.
    
  *  If F ∈𝒱∖ i, then δ̅^*(y_F) = x_F ∪ i.
    
  *  The pullback of the special element y_i = -∑_F ∈𝒱∖ i y_F is
    δ̅^*(y_i) = δ̅^*(-∑_F ∈𝒱∖ i y_F) = -∑_F ∈𝒱 x_F = ∑_F ∈𝒰 x_F.


    𝒱
    𝒯      𝒰
       𝒮∪{∅}[from=2-1, to=1-2]
    	[from=2-3, to=1-2]
    	[from=3-2, to=2-1]
    	[from=3-2, to=2-3]


For ranks r=1, …, d+1, let 

    t_r := ∑_F ∈𝒯
    (F) = r x_F,      v_r := ∑_F ∈𝒱
    (F) = r x_F,
    
        s_r := ∑_F ∈𝒮
    (F) = r x_F,       u_r := ∑_F ∈𝒰
    (F) = r x_F.

The one piece may be empty in some rank r, in which case the sum is zero. For short hand, we also group indices with the same rank (t + s)_r := t_r + s_r, etc.

Using this notation,

    (M)    = [1 - (t + s + u + v)_1 - (t + s + u + v)_2 - … - (t + s + u + v)_d+1]
           · [1 - (t + s + u + v)_2 - … - (t + s + u + v)_d+1]
           ⋯
           · [1 - (t + s + u + v)_d+1]

and

    δ̅^*((M∖ i))    = [1 - (t_1 + (s_1 + u_2) + v_1) - (t_2 + (s_2 + u_3) + v_2) - … - (t_d+1 + (s_d+1 + u_d+2) + v_d+1)]
           · [1 - (t_2 + (s_2 + u_3) + v_2) - (t_3 + (s_3 + u_4) + v_3) - … - (t_d+1 + (s_d+1 + u_d+2) + v_d+1)]
           ⋯
           · [1 - (t_d+1 + (s_d+1 + u_d+2) + v_d+1)]
       = [1 - (t + s + v)_1 - (t + s + u + v)_2 - … - (t + s + u + v)_d+1]
           · [1 - (t + s + v)_2 - (t + s + u + v)_3 - … - (t + s + u + v)_d+1]
           ⋯
           · [1 - (t + s + v)_d+1]


Note u_r is omitted from the i-th factor. We compute the difference (M) - δ̅^*((M∖ i)) and show it is equal to the negative of


    δ̅^* (y_i ·ι((M/i)))=
       [u_1 + u_2 + … + u_d+1] 
         ·[1 - (s_1 + u_2 + v_2) - (s_2 + u_3 + v_3) - … - (s_d + u_d+1 + v_d+1)]
         ·[1 - (s_2 + u_3 + v_3) - … - (s_d + u_d+1 + v_d+1)]
         ⋯
         · [1 - (s_d + v_d+1 + u_d+1)]


The difference is:


    (M) - δ̅^*((M∖ i)) =
       [-u_1]·[1 - (t + s + u + v)_2 - … - (t + s + u + v)_d+1] ⋯ [1 - (t + s + u + v)_d+1]
       +[1 - (t + s + v)_1 - (t + s + u + v)_2 - … - (t + s + u + v)_d+1] · [-u_2] ⋯ [1 - (t + s + u + v)_d+1]
       + ⋯
       +[1 - (t + s + v)_1 - … - (t + s + u + v)_d+1] · [1 - (t + s + v)_2 - … - (t + s + u + v)_d+1] ⋯ [-u_d+1]


Note in the r-th term is like (M) except the r-th factor is replaced by -u_r and for each k < r, u_k each omitted from the k-th factor. Each flat in 𝒯 is not comparable with each flat in 𝒰, so t_k u_r = 0. In other words, t_k is annihilated by each u_r. Since each term has some u_r as a factor, all `t's may be removed.


    (M) - δ̅^*((M∖ i)) = … =
       [-u_1]·[1 - (s + u + v)_2 - … - (s + u + v)_d+1] ⋯ [1 - (s + u + v)_d+1]
       +[1 - (s + v)_1 - (s + u + v)_2 - … - (s + u + v)_d+1] · [-u_2] ⋯ [1 - (s + u + v)_d+1]
       + ⋯
       +[1 - (s + v)_1 - … - (s + u + v)_d+1] · [1 - (s + v)_2 - … - (s + u + v)_d+1] ⋯ [-u_d+1]


Also u_r s_k = 0 if k ≥ r. Then, for each term, index r, for all k>r add s_k-1 to the k-th factor (after each [-u_r]).


    (M) - δ̅^*((M∖ i)) =… = 
       [-u_1]·[1 - s_1 - (s + u + v)_2 - … - (s + u + v)_d+1] ⋯ [1 - s_d - (s + u + v)_d+1]
       +[1 - (s + v)_1 - (s + u + v)_2 - … - (s + u + v)_d+1] · [-u_2] ⋯ [1 - s_d - (s + u + v)_d+1]
       + ⋯
       +[1 - (s + v)_1 - … - (s + u + v)_d+1] · [1 - (s + v)_2 - … - (s + u + v)_d+1] ⋯ [-u_d+1]


Also u_r v_k = 0 if k < r. Then, for each term, index r, for all k < r remove v_k from the k-th factor (before each [-u_k]).

    (M) - δ̅^*((M∖ i)) =… = 
       [-u_1]·[1 - s_1 - (s + u + v)_2 - … - (s + u + v)_d+1] ⋯ [1 - s_d - (s + u + v)_d+1]
       +[1 - s_1 - (s + u + v)_2 - … - (s + u + v)_d+1] · [-u_2] ⋯ [1 - s_d - (s + u + v)_d+1]
       + ⋯
       +[1 - s_1 - … - (s + u + v)_d+1] · [1 - s_2 - … - (s + u + v)_d+1] ⋯ [-u_d+1]

Factors before and after u_r are identical between terms. Factorize:

    (M) - δ̅^*((M∖ i)) =… = 
       [u_1 + u_2 + … + u_d+1] 
         ·[1 - s_1 - (s + u + v)_2 - (s + u + v)_3 - … - (s + u + v)_d+1]
         ·[1 - s_2 - (s + u + v)_3 - … - (s + u + v)_d+1]
         ⋯
         · [1 - s_d - (s + u + v)_d+1]
s_d+1 = 0, since x_E is the only flat of rank d+1, and i ∈ E. That is, E ∉𝒮.

    (M) - δ̅^*((M∖ i)) =… = 
       -[u_1 + u_2 + … + u_d+1] 
         ·[1 - s_1 - (s + u + v)_2 - (s + u + v)_3 - … - (u + v)_d+1]
         ·[1 - s_2 - (s + u + v)_3 - … - (u + v)_d+1]
         ⋯
         · [1 - s_d - (u + v)_d+1]
       =
       -[u_1 + u_2 + … + u_d+1] 
         ·[1 - (s_1 + u_2 + v_2) - (s_2 + u_3 + v_3) - … - (s_d + u_d+1 + v_d+1)]
         ·[1 - (s_2 + u_3 + v_3) - … - (s_d + u_d+1 + v_d+1)]
         ⋯
         · [1 - (s_d + u_d+1 + v_d+1)]
       = -δ̅^* (y_i ·ι((M/i)))

As desired, we have

    (M) = δ̅^*(M∖ i) - δ̅^* (y_i ·ι((M/i)))



























































 §.§ Piecewise polynomials and cocycles


There is an equivalent formulation ofA^*(Σ)in terms of so-called Courant functions onΣ. This approach definesA^*(Σ)as a quotient of continuous piecewise polynomial functions onΣby linear functions fromNtoℝ, see Section 4 of <cit.> for more details. 

This other formulation leads to the theory of tropical cocycles onΣ, introduced by François in <cit.> and elaborated upon by Gross and Shokrieh in <cit.>, in analogy to the theory of cycles onΣfrom <cit.>. In particular,Z_*(Σ), the tropical cycles onΣ, consists of weighted fans on subdivisions ofΣ, whileC^*(Σ), the tropical cocycles onΣ, are continuous functions that are piecewise polynomial in a subdivision ofΣ. In <cit.>, the authors prove that, when given the corresponding intersection products, theseC^*(Σ)-algebras are isomorphic. In particular, given a morphism of fansf:Σ_1 →Σ_2, a cocycleh ∈C^*(Σ_2)and a cycleX ∈Z_*(Σ_2),f^*(h ·X) = f^*(h) ·f^*(X), wheref^*:Z_*(Σ_2) →Z_*(Σ_1)is defined in Corollary 4.7 in <cit.>, which agrees with Definition 2.16 in <cit.> in the case wherefis a tropical modification. 

Using this theory, we describe the pullbackδ^*:Z_*(B(M ∖i)) →Z_*(B(M))in terms of cocycles, for a matroidMof rankron a setEandi ∈Enot a coloop. Since in our case the mapδ:B(M) →B(M ∖i)maps cones to cones, the image of weights onB(M ∖i)underδ^*is contained in weights onB(M). Similarly, the image underδ^*of a piecewise polynomial onB(M ∖i)is a piecewise polynomial onB(M), giving a pullbackA^*(M ∖i ) →A^*(M). 

Sinceδ̂:B̂(M) →B̂(M ∖i)satisfies the conditions of Proposition 3.10 in <cit.>, there exists a rational functionϕsuch thatδis a tropical modification alongϕ, and there is an isomorphismϕ·MW_*(B̂(M ∖i))  MW_*(B̂(M / i)). The inverse of this isomorphism is given byĵ_*:MW_*(B̂(M / i)) →MW_*(B̂(M ∖i)).


    The inclusion j_*:MW_*(B(M / i)) → MW_*(B(M ∖ i)) is given by j_*(z ∩ 1_M / i) = y_i ·ι(z) ∩ 1_M ∖ i.

    By Proposition 3.10 in <cit.>, ϕ is given by 
    
    ϕ(v_σ̂_F) =    _M / i(F) -_M ∖ i(F) 
    
            =    _M(F ∪ i) - _M(i) - _M(F) 
    
            =    _M(F ∪ i) - _M(F) - 1.

    If F is not a flat in M, but F ∪ i is, then _M(F ∪ i) = _M(F), thus
    
    ϕ(v_σ̂_F) =    _M(F ∪ i) - _M(F) - 1 
    
            =    _M(F) - _M(F) - 1 
    
            =     -1.

    If F is a flat in M, then _M(F ∪ i) = _M(F) + 1, thus
    
    ϕ(v_σ̂_F) =    _M(F ∪ i) - _M(F) - 1 
    
            =    _M(F) + 1 - _M(F) - 1 
    
            =     0.

    Therefore, under the isomorphism MW_*(B̂(M)) ≅ MW_*(B(M)), ϕ·_ = y_i ∩_, where y_i is defined in (<ref>). Finally, it is straightforward check to see that y_i ·ι(z) ∩ 1_M ∖ i = j_*(z ∩ 1_M / i).

    For cocycles y ∈ A^*(M∖ i ), δ̅^*(y) ∩ 1_M = δ^*(y ∩ 1_M ∖ i).

    Since these are ring homomorphisms and both rings are generated in degree 1, we need only check this holds for Courant functions. Courant functions are uniquely determined by their values on rays; thus the claim simplifies to the statement that for all flats F ∈ℒ̂(M ∖ i), if F is a flat in M, then δ^-1(σ_F) = σ_F∪σ_F ∪ i and x_F(e̅_F) = x_F(δ(e_F∪ i)) = x_F(δ(e_F)), and if F is not a flat of M then δ^-1(σ_F) = σ_F ∪ i and x_F(e̅_F) = x_F(δ(e_F∪ i)). From the definition of δ, it is clear that δ(e_F∪ i) = δ(e_F) = e̅_F, thus proving the claim. 

    For a matroid M, (M) ∩ 1_M = (M).

    We proceed by induction. Note that for the loop matroid M_0, i.e. the matroid on one element whose only element is a loop, (M_0) = 0 = 0 ∩ 1_M_0 = (M_0) ∩ 1_M_0 and for the isthmus M_1, i.e. the matroid on one element whose only element is independent, (M_1) = 1_M_1. Now, assume that for all matroids M' whose ground set consists of n elements (M') ∩ 1_M' = _k(M') and let M be a loopless matroid whose ground set consists of n + 1 elements. If all elements of the ground set of M are coloops, then M is representable over ℂ and thus, by Theorem <ref>, the conclusion follows. Now assume there exists i ∈ M that is not a coloop. Then, by Theorem <ref>, (M) = δ̅^* (M ∖ i) - δ̅^* (M / i). Then, the ground sets of M ∖ i and M / i contain n elements. Thus, by Lemma <ref> and Theorem 5.4 in <cit.>,
    
    (M) ∩ 1_M =    
    
            =    δ̅^* ((M ∖ i)) ∩ 1_M - δ̅^* (y_i ·ι ((M / i))) ∩ 1_M
    
            =    δ^* ((M ∖ i) ∩ 1_M ∖ i) - δ^*( y_i ·ι ((M / i)) ∩ 1_M ∖ i)  
    
            =    δ^*(M ∖ i) - δ^* j_*((M / i) ∩ 1_M / i) 
    
            =    δ^*(M ∖ i) - δ^* j_*(M / i) 
    
            =    (M).


Thus, we obtain as a consequence of the computation in Theorem <ref> and Theorem <ref>:


    The formula of Fife and Rincón holds for all matroids.
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