On tropical intersection theory

We develop a tropical intersection formalism of forms and currents that extends classical tropical intersection theory in two ways. First, it allows to work with arbitrary polyhedra, also non-rational ones. Second, it allows for smooth differential forms as coefficients. The intersection product in our formalism can be defined through the diagonal intersection method of Allermann–Rau or the fan displacement rule. We prove with a limiting argument that both definitions agree.


Introduction
For their "Tropical approach to non-archimedean Arakelov theory" [9], Gubler-Künnemann combine tropical intersection theory and smooth differential forms into their formalism of so-called δ-forms.They use these to develop a calculus of Green currents on non-archimedean spaces that is related to intersection theory on formal models.The strength of their approach is that δ-forms are simpler to work with than formal models, leading to a computationally accessible handle for certain arithmetic intersection problems.
The present paper contributes to these ideas through the development of a more general and concise theory of δ-forms.This is a purely tropical endeavor: δ-Forms are a natural generalization of tropical cycles and have the same formal properties.For example, they admit pull-backs, push-forwards and a tropical intersection product called the ∧-product.δ-Forms also encompass Lagerberg's smooth forms [11] and obey the same kind of differential calculus.They furthermore come with a boundary operator that generalizes the frequently used corner locus constructions of Esterov [5] and Francois [6], also cf.Gubler-Künnemann [9].Moreover, our formalism allows nonrational polyhedra throughout.For tropical cycles, this generalization had already been obtained by Esterov [5].
We now provide a more detailed description of δ-forms and our results.Smooth forms are always meant in the sense of Lagerberg in the following, cf.[11] or §2.1.Recall that a current is a continuous linear form on the space of smooth forms with compact support.A smooth form α on R n and a polyhedron σ ⊆ R n define a current of integration (α∧σ)(η) := σ α∧η.(The polyhedron really needs to be weighted for this to work which will be explained below.)A current is called polyhedral if it is a locally finite sum i∈I α i ∧ σ i of such integration currents.In particular, polyhedral currents are entirely combinatorial objects.The following is our main definition.Definition 1.1.A δ-form on R n is a polyhedral current T on R n such that both derivatives d ′ T and d ′′ T are again polyhedral.
The differentials d ′ and d ′′ here are taken in the sense of currents, i.e. as the duals of d ′ and d ′′ for smooth forms.δ-Forms turn out to be stable under d ′ and d ′′ .Additional structure is then provided by defining a δ-form T = i∈I α i ∧ σ i to be of tridegree (p, q, r) if the α i may be chosen of bidegree (p, q) and the σ i of codimension r.Then d ′ naturally decomposes as d ′ = d ′ P − ∂ ′ , where d ′ P is trihomogeneous of tridegree (1, 0, 0) and ∂ ′ trihomogeneous of tridegree (0, −1, 1).The first summand d ′ P is the so-called polyhedral derivative d ′ P (α ∧ σ) = (d ′ α) ∧ σ of Gubler-Künnemann, while ∂ ′ is the above-mentioned boundary operator.The latter is closely related to boundary integration of differential forms and to the corner locus construction, cf.(4.13).A similar decomposition d ′′ = d ′′ P − ∂ ′′ exists for d ′′ .Next, we come to the combinatorial description of δ-forms.
Theorem 1.2.A polyhedral current T = i∈I α i ∧ σ i is a δ-form if and only if the datum (α i , σ i ) i∈I is balanced in the sense of tropical geometry.
We formulate the relevant balancing condition in (1.1) below.Note that Thm.1.2 has precursors in the literature: Lagerberg [11,Prop. 4.7], Gubler [8,Prop. 3.8] and Prop. 2.16] (in successive level of generality) essentially prove it whenever the α i are smooth functions.Cast in our terminology, they show that the tropical cycles of codimension r with smooth coefficients are exactly the δ-forms of tridegree (0, 0, r).
Thm. 1.2 makes δ-forms behave like tropical cycles and we show that Allermann-Rau's construction of an intersection product [1] goes through without substantial change.This leads to our main result which is clearly inspired by Gubler-Künnemann's [9,Prop. 4.15].

Theorem 1.3.
There is a graded-commutative ∧-product of δ-forms that extends the ∧-product of smooth forms and the intersection product of tropical cycles.The derivatives d ′ , d ′′ , the polyhedral derivatives d ′ P , d ′′ P and the boundary derivatives ∂ ′ , ∂ ′′ all satisfy the Leibniz rule for ∧.A more precise characterization of the ∧-product may be found in the main text, cf.Thm.4.1.We also show that the ∧-product can be computed by the fan displacement rule, cf.Prop.4.21.Recall that for intersections of tropical cycles, this rule goes back to Fulton-Sturmfels [7] and Mikhalkin [12].Its equality with Allermann-Rau's intersection product was shown independently by Rau [14] and Katz [10].Our proof is similar to the combinatorial one of Rau and based on the observation that the ∧-product suitably commutes with limits, cf.§4. 3.
We next explain the tropical formalism for possibly non-rational polyhedra.For a polyhedron σ ⊆ R n , denote by N σ ⊆ R n the linear space spanned by all x − y, x, y ∈ σ.Given a facet τ ⊂ σ, the subspace N τ ⊂ N σ is of codimension 1.If σ is rational, then (N τ ∩ Z n ) ⊂ (N σ ∩ Z n ) is a sublattice of corank 1 and a normal vector for τ ⊂ σ is any vector n σ,τ ∈ N σ ∩ Z n that generates (N σ ∩ Z n )/(N τ ∩ Z n ) and points in direction of σ.For the general situation, we consider weighted polyhedra instead.A weight for σ is simply a generator µ σ ∈ det N σ up to sign.Equivalently, it is a choice of Haar measure on N σ .Given a facet inclusion τ ⊂ σ and respective weights µ τ and µ σ , a normal vector is any n σ,τ ∈ N σ that satisfies µ σ = µ τ ∧ n σ,τ and points in direction of σ.The two definitions are linked by the observation that every rational polyhedron σ has a natural weight, namely the unique-up-to-sign generator of det Z (N σ ∩ Z n ).The balancing condition (1.1) in Thm.1.2 is now a literal adaption of the classical balancing condition.Definition 1.4.Consider a polyhedral complex T , weights (µ σ ) σ∈T for its polyhedra and smooth forms (α σ ) σ∈T , α σ ∈ A(σ).Here, A(σ) denotes the smooth forms on σ.This datum is called balanced if for all τ ∈ T , σ∈T , τ ⊂σ a facet We next elucidate on the intersection theory of weighted polyhedra.Recall that given two properly intersecting rationally defined subspaces N 1 , N 2 ⊆ R n , one defines their intersection multiplicity as the lattice index In the not necessarily rational case, still assuming proper intersection, one instead considers weights µ 1 , µ 2 for N 1 , N 2 and endows the intersection N 1 ∩ N 2 with the unique weight ν such that Here µ std is the standard weight on R n .This rule extends to a full description of the ∧-product of transversally intersecting δ-forms and underlies the fan displacement rule.
Finally, a weight µ for σ is also the precise datum needed to define the integral [σ,µ] η of a (compactly supported) form η over σ.So in the definition of polyhedral current above, all polyhedra were silently weighted.For this natural reason, weights implicitly occur in Lagerberg [11] and Chambert-Loir-Ducros [3].In fact, the calibrages from [3] are the same as our weights with an additional sign.
Tropical intersection theory has also been extended from R n to more general combinatorial spaces.We will not address such questions here but take them up in our related work [13].More precisely, we develop there a theory of δ-forms on so-called tropical spaces with applications to non-archimedean Arakelov theory.
Layout §2 contains a summary of Lagerberg's theory of differential forms and introduces the formalism of weights, normal vectors and fiber integration.§3 is dedicated to the definition of δ-forms and to the proof of Thm.1.2.§4 contains the main result Thm.1.3 and some additional properties of δ-forms.The fan displacement rule is Prop.4.21 and will be proved in §4.3.
There is a bigrading A = p,q A p,q , where A p,q is the piece Ω p ⊗ C ∞ Ω q .Elements α ∈ A p,q are called bihomogeneous of bidegree (p, q) and homogeneous of degree deg α = p + q.
Being an exterior algebra, A is endowed with a natural ∧-product.It is bihomogeneous in the sense that A p,q ∧ A s,t ⊆ A p+s,q+t .It is also graded-commutative, meaning that whenever α and β are homogeneous.We use the terminology of [3] for differential operators.Write d std : (2.3) Denoting by x 1 , . . ., x n the standard coordinates on R n , any α ∈ A p,q is now in a unique way of the form α = I,J⊆{1,...,n}, |I|=p, |J|=q Concretely, this extension is given as The so-defined d ′ , d ′′ : A → A are bihomogeneous of bidegree (1, 0) resp.(0, 1).
Given an affine-linear map f : R n → R m , there is a pull-back map f * : A p,q (R m ) → A p,q (R n ) which stems from usual pull-back of differential forms.It commutes with ∧, d ′ and d ′′ .
The integral of an (n, n)-form η with compact support is defined as follows.Write where the right hand side is defined in terms of the Lebesgue integral for the standard volume on R n .It is immediate that, for an affine linear map f : There is, in particular, no implicit choice of orientation involved.This also reflects in the fact that the forms d ′ x i ∧ d ′′ x i have degree 2, hence pairwise commute, so the function ϕ in (2.5) is independent of coordinate ordering.Let D = D(R n ) denote the space of currents, i.e. the topological dual of compactly supported forms A c , cf. [11, §1.1].The topological aspect of the definition will never play a role in this paper.Write D = p,q D p,q where D p,q is dual to A n−p,n−q c .There is an injective map A p,q → D p,q , α → [α], defined by With the following sign conventions one defines derivatives d ′ : D p,q → D p+1,q , d ′′ : D p,q → D p,q+1 as well as a product ∧ : A p,q × D s,t → D p+s,q+t : (2.7) Then it follows that, for homogeneous α, β, T and d ∈ {d ′ , d ′′ }, for every compactly supported test form η, so the inclusion A → D commutes with ∧, d ′ and d ′′ .Furthermore, the Leibniz rule extends: (2.9) Let f : R n → R m be an affine linear map.Then there is a push-forward map D p,q c (R n ) → D p+m−n,q+m−n (R m ) from currents with compact support.It is defined by (2.10)

Polyhedral currents
By polyhedron in R n we mean a subset σ that may be written as the intersection of finitely many (not necessarily rational) half-spaces.Denote by N σ the linear space spanned by all x − y, x, y ∈ σ and by M σ := N ∨ σ its R-dual.The dimension of σ is the dimension of N σ .
A polyhedral complex is a locally finite set of polyhedra T which is stable under taking faces and is such that σ 1 ∩ σ 2 is a face of both σ 1 and σ 2 for every σ 1 , σ 2 ∈ T .The d-dimensional (resp.r-codimensional in R n ) polyhedra of a polyhedral complex are denoted by T d (resp.T r ).
Let C ∞ (σ) denote the smooth functions on σ, i.e. all ϕ : σ → R such that there is some smooth function ϕ ∈ C ∞ (R n ) with ϕ| σ = ϕ.Smooth (p, q)-forms on σ are defined by an analogous restriction process.Let L σ = x + N σ , x ∈ σ, be the smallest affine linear space containing σ.There is a well-defined space of (p, q)-forms A p,q (L σ ) because (p, q)-forms transform naturally under affine linear maps.Then Equivalently, it is the space of smooth (p, q)-forms on the interior σ • of σ in L σ that come by restriction from A p,q (L σ ).Note that A p,q (σ) = 0 if dim σ < max{p, q} and that there is a restriction map A p,q (σ) → A p,q (τ ), α → α| τ for every inclusion of polyhedra τ ⊆ σ which commutes with ∧, d ′ and d ′′ .The following definition of weight is inspired by Chambert-Loir-Ducros' definition of a calibration, cf.[3, §1.5].
Definition 2.1.A weight on a polyhedron σ is a generator µ ∈ det N σ up to sign.The convention for 0-dimensional polyhedra here is that the determinant of the 0-space is R itself and that a weight is a positive scalar.The pair [σ, µ] is called a weighted polyhedron.A weighted polyhedral complex is the datum of a polyhedral complex T together with weights (µ σ ) σ∈T for all its polyhedra.Equivalently, a weight is the choice of a Haar measure for N σ .We denote by µ ∨ ∈ det M σ the dual of µ.
Example 2.2.Every rational polyhedron σ ⊆ R n has a natural weight with respect to the lattice Z n ⊆ R n .Namely N σ ∩ Z n is a lattice in N σ and the choice of a generator µ 0 ∈ det Z (N σ ∩ Z n ) is unique up to sign.Every other weight is in a unique way of the form λµ 0 , λ > 0.

Let [σ, µ] be a weighted polyhedron of dimension d and let
(The x i are defined up to translation on L σ , so their differentials d ′ x i and d ′′ x i are canonical.)Then set where the right hand side is the Lebesgue integral with respect to the volume defined by the choice of isomorphism (x 1 , . . ., x d ) : N σ ∼ = R d .The transformation rule (2.6) ensures that this is well-defined.In this way, [σ, µ] is viewed as element of D r,r , where r = n − d is the codimension of σ.
The following definitions are due to Gubler-Künnemann, cf.[9, Def.2.3].A polyhedral current is a current that is a locally finite sum of currents of the form α ∧ [σ, µ].Deviating from their notation, we write P ⊆ D for the space of all polyhedral currents and P p,q,r ⊆ D p+r,q+r for those which are locally finite sums of α ∧ [σ, µ] with σ of codimension r and α ∈ A p,q (σ).One easily checks the direct sum decomposition P = p,q,r P p,q,r .We also say that elements of P p,q,r are trihomogeneous of tridegree (p, q, r).
Remark 2.3.When presenting a polyhedral current T as a locally finite sum T = i∈I α i ∧ [σ i , µ i ], the datum of all (α i , σ i , µ i ) i∈I is unique up to locally finitely many operations of the following kinds: Subdividing the σ i , adding/removing terms with α = 0, replacing (α, σ, µ) by (λα, σ, λ −1 µ) for some λ > 0, and exchanging Its polyhedral derivatives are the polyhedral currents It has been remarked before, cf.[9, Rmk.2.4 (iii)], that d ′ P T and d ′ T resp.d ′′ P T and d ′′ T need not coincide.The derivatives d ′ T and d ′′ T may even be non-polyhedral, cf.Ex. 2.10 below.
A polyhedral complex T is subordinate to T if there is a presentation of the form T = With such T fixed, the α σ and µ σ are uniquely determined up to the replacement of (α σ , µ σ ) by (λα σ , λ −1 µ σ ), where λ > 0.

Functoriality
For an exact sequence of finite dimensional R-vector spaces there is a canonical-up-to-sign isomorphism det N 2 = det N 1 ⊗ det N 3 .So given weights µ i for N i for two out of {N 1 , N 2 , N 3 }, they uniquely determine a weight for the third space through the relation where There is a space P S(σ) of piecewise smooth forms on a polyhedron σ.By definition, a piecewise smooth form is the datum of a polyhedral complex T with σ = ∪ ρ∈T ρ and smooth forms α ρ ∈ A(ρ), ρ ∈ T , such that α ρ | τ = α τ for all τ ⊆ ρ; up to subdivision.We write P S p,q (σ) for those with all α ρ of bidegree (p, q).If µ is a weight for σ and α = (α ρ ) ρ∈T ∈ P S(σ) as before, we define the polyhedral current Here µ defines a weight for ρ because N ρ = N σ for dimension reasons.For fixed µ, this defines an embedding P S(σ) ⊆ D(R n ).
Let f : R n → R m be an affine linear map and σ ⊆ R n a polyhedron.Then f (σ) is again a polyhedron.Let µ be a weight on σ and ν a weight on f (σ).
12) and there is a natural fiber integration map for forms with compact support, f δ, * : which determines it uniquely.In other words, fiber integration provides a representative for the push-forward from (2.10), In particular, the push-forward of a polyhedral current (with relatively compact support) is polyhedral again.Note that if α ∧ [σ, µ] ∈ P p,q,r c and dim K = k as before, then f * (α ∧ [σ, µ]) ∈ P p−k,q−k,r+m−n+k .So f * is not trihomogeneous, but only bihomogeneous.
which is precisely the classical push-forward of weighted polyhedra in tropical geometry that underlies e.g. the Sturmfels-Tevelev multiplicity formula [15].Given a surjective affine linear map f : R n → R m and a current T on R m , we may now also define a pull-back current f * T ∈ D(R n ).Namely the fiber integral f * η of a smooth form η ∈ A p,q c (R n ) (with respect to the standard weights on R n and R m ) is again smooth and we put Example 2.6.Assume f : R n → R n is bijective and affine linear.Let µ be the standard weight on R n .Then The transformation rule (2.6) implies that this satisfies the projection formula: Interchanging the roles of α and η, the equality of leftmost and rightmost term shows (2.16)

Stokes' Theorem
By its very definition, a (p, q)-form α on R n may be viewed as an alternating form in p+q variables on R n × R n with values in C ∞ .Given w ∈ R n × R n , the contraction (α, w) of α with w (interior derivative) is defined as the multilinear form resulting from inserting and fixing w as the first entry of α.This operation is characterized by the Leibniz rule and the identities Recall that a facet of a polyhedron is a face of codimension 1.
Definition 2.7.Let [σ, µ] be a weighted polyhedron and τ ⊂ σ a facet that is endowed with a weight ν.Then there is a unique vector n σ,τ ∈ N σ /N τ that points in direction of σ and is such that µ = ν ∧ n σ,τ in the sense of (2.12).A normal vector for τ ⊂ σ is any choice of lift The convention in notation here is v ′ = (v, 0) and v ′′ = (0, v) for any vector v ∈ R n .The restriction (α, n ′′ σ,τ )| τ is independent of the choice of normal vector and the whole expression is independent of the choice of ν.Define the (first) boundary integral of σ as α. (2.20) The definition of the (second) boundary integral differs by a sign which is motivated by Ex. 2.9 below.For Example 2.9.Prop.2.8 is essentially just the following statement.For every smooth function The differing signs explain the sign change from (2.19) to (2.21).
as currents, but these derivatives are never polyhedral if dim σ > 0. (The difference in sign with Stokes' Theorem comes from (2.7).)Namely they have support on the union of facets ∂σ of σ.

δ-Forms
We consider forms, currents and polyhedra on R n in the following.
Definition 3.1.A δ-form is a polyhedral current T such that both d ′ T and d ′′ T are again polyhedral.
This definition turns out to be equivalent to the familiar concept of balancing for T .
Definition 3.2.Let T be a polyhedral complex, µ σ , σ ∈ T , a family of weights for its polyhedra and α σ ∈ A(σ), σ ∈ T a family of smooth forms.This datum is called balanced, if the following two equivalent conditions are met.
(2) For every polyhedron τ ∈ T , every affine linear function z with constant restriction z| τ and normal vectors n σ,τ as before, Since z| τ is constant, this expression does not depend on the choices of the n σ,τ .
Proof of the equivalence of ( 1) and ( 2).Assume z| τ to be constant and consider the pairing It follows from the Leibniz rule that so the pairing factors through A(τ ) ⊗ R R n and is simply the A(τ )-linear extension of v → ∂z/∂v.The proof is now the statement that a vector v lies in N τ if and only if ∂z/∂v = 0 for every affine linear function z with constant restriction z| τ .
Formulation (1) is closer to the usual condition of balancing in tropical geometry but makes implicit use of the existence of the ambient space R n .Formulation (2) in turn is more suitable for generalizations to abstract polyhedral complexes, cf.[13].
Being balanced is stable under the four operations in Rmk.2.3, so only depends on the current T = σ∈T α σ ∧ [σ, µ σ ].Also note that (3.1) and (3.2) are trihomogeneous in α and that only polyhedra of a fixed dimension occur.One obtains that T = p,q,r T p,q,r , with T p,q,r of tridegree (p, q, r), is balanced if and only if each T p,q,r is.Theorem 3.3.A polyhedral current T is a δ-form if and only if it is balanced.In particular, it is a δ-form if and only if T p,q,r is a δ-form for all (p, q, r).
Furthermore, T is already a δ-form if one out of d ′ T , d ′′ T is again polyhedral. Proof.
(1) We first assume that T is of tridegree (p, q, r).Let T be a weighted polyhedral complex subordinate to T , say and let η ∈ A n−p−r,n−q−r c be a test form.One obtains from the Leibniz rule and Stokes' Theorem that The individual contractions (α σ , n ′′ σ,τ ) and (η, n ′′ σ,τ ) depend on the choices of normal vectors, but the total expression does not.We henceforth fix the choices n σ,τ .The terms (α σ , n ′′ σ,τ )∧[τ, µ τ ] (η) always define polyhedral currents.So the statement to prove is that T is balanced if and only if the following is a polyhedral current, (2) Assume first that T is balanced, fix some τ and write according to (3.1).Then By the Leibniz rule, Since β i ∧ η is of bidegree (dim τ, dim τ + 1), the first summand vanishes.The (integral over [τ, µ τ ] of the) second summand defines a polyhedral current in η.Taking the sum over i and τ shows that d ′ T is a polyhedral current.The same argument works for d ′′ T , proving that a trihomogeneous balanced polyhedral current is a δ-form.
(3) Conversely assume that T is not balanced, our claim being that d ′ T is not polyhedral.(We still assume that T has tridegree (p, q, r) currently.)Generally, if S is a polyhedral current, C some polyhedral sets with Supp S ⊆ C and η ∈ A c a test form, then η| C = 0 implies S(η) = 0.In the situation at hand, we have already seen that Supp(d ′ T − d ′ P T ) is contained in the codimension r + 1 skeleton τ ∈T r+1 τ and our approach is to construct a test form η with η| τ = 0 for all τ but (d ′ T − d ′ P T )(η) = 0. Pick τ and z such that (3.2) is not satisfied, i.e. z is an affine linear function with constant restriction z| τ and such that There exists a bump test form η ∈ A dim τ −p,dim τ −q c with the properties that Supp η ∩ τ ′ = ∅, τ ′ ∈ T r+1 , only for τ ′ = τ and Then the τ -contribution to (3.3) for the test form η = (−1) Here we combined the Leibniz rule for ( , n ′′ σ,τ ) with the properties d ′′ z| τ = 0 and . So d ′ T cannot be polyhedral and hence T is not a δ-form.Note that arguments (2) and (3) show the stronger statement that a trihomogeneous T is balanced if and only if one out of d ′ T and d ′′ T is polyhedral, i.e. they prove the last statement of Thm.3.3 for trihomogeneous T .
(4) Now consider a general δ-form T = p,q,r T p,q,r with T p,q,r of the indicated tridegree.Our claim is that each T p,q,r is a δ-form.Since T p1,q1,r and T p2,q2,r lie in different bidegrees as currents for (p 1 , q 1 ) = (p 2 , q 2 ) and since d ′ and d ′′ are bihomogeneous, it is enough to prove that all T r := p,q T p,q,r are δ-forms.Polyhedral currents may be added ad libitum, so it is sufficient to show that all (d ′ T r − d ′ P T r ) and (d ′′ T r − d ′′ P T r ) are polyhedral.Assume for the sake of contradiction that there is some r 0 with d ′ T r0 not polyhedral and assume that r 0 is chosen minimal.Then the previous arguments imply that there is some point that has no open neighborhood x ∈ U such that d ′ T r0 | U is polyhedral.(Take x ∈ Supp β where β is as in (3.7).)Using minimality of r 0 , we conclude that d ′ T cannot be polyhedral.The same argument applies with d ′′ instead of d ′ .Thus we obtain that T is a δ-form, if and only if each T p,q,r is a δ-form, if and only if each T p,q,r is balanced, if and only if T is balanced.
(1) We denote by B p,q,r = B p,q,r (R n ) the space of δ-forms of the indicated tridegree and by B = p,q,r B p,q,r the space of all δ-forms.Write B p,q = r B p−r,q−r,r for the space of δ-forms of bidegree (p, q) in the sense of currents. ( Similarly for d ′′ , so one obtains derivatives d ′ : B p,q −→ B p+1,q , d ′′ : B p,q −→ B p,q+1 . The balancing condition (3.1) is stable under d ′ P and d ′′ P , so the polyhedral derivatives restrict to operators d ′ P : B p,q,r −→ B p+1,q,r , d ′′ P : B p,q,r −→ B p,q+1,r .Define the boundary operators It will be explained below, cf.(3.9), that these are trihomogeneous in the sense ∂ ′ : B p,q,r −→ B p,q−1,r+1 , ∂ ′′ : B p,q,r −→ B p−1,q,r+1 .Lemma 3.5.(1) The boundary derivatives satisfy (2) Assume that T ∈ B c (R n ) has compact support and that f : R n → R m is an affine linear map. Then ) is also a δ-form.
(3) Let f : R n → R m be a surjective affine linear map and S ∈ B(R m ).Then f * S ∈ B(R n ) is also a δ-form. Proof.
(1) The necessary observation is that d ′ P , ∂ ′ , d ′′ P and ∂ ′′ are all trihomogeneous of different tridegrees.The stated relations then follow from the identities (2) and (3) follow from the fact that f * and f * commute with d ′ and d ′′ and preserve the property of being polyhedral.
Lemma 3.7.The δ-forms B p,q,0 (R n ) are precisely the currents of the form α ∧ [R n , µ std ] for a piecewise smooth (p, q)-form α ∈ P S p,q (R n ).
Example 3.8.Let α ∈ P S(R n ) be piecewise smooth and T ∈ P (R n ) a polyhedral current.Let T be a polyhedral complex that is subordinate to both T and α, say Define their product as αT := The restriction α| ρ here is well-defined by the piecewise smooth property.If T is a δ-form, then αT is also a δ-form since the balancing condition (3.1) is P S-linear.For example, the δ-preforms from [9, §2] are precisely the sums of products αT where α ∈ A(R n ) is smooth and T ∈ B 0,0,r a tropical cycle.
We end this section by providing three ways to compute ∂ ′ T .The case of ∂ ′′ is the same by symmetry; it merely requires paying attention to difference in signs of (2.19) and (2.21).Throughout, we assume that T ∈ B p,q,r , say T = σ∈T r α σ ∧ [σ, µ σ ] for a subordinate weighted polyhedral complex T .The proof of Thm.3.3 shows that ∂ ′ T = τ ∈T r+1 β τ ∧ [τ, µ τ ] for certain β τ which we would like to determine.
(1) Implicit in the proof of Thm.3.3 is the following formula.Fix τ and write as in (3.4).Then (3.3), together with (3.5) and (3.6), implies (2) The next formula for β τ is more in line with formulation (3.2) of the balancing condition.We use the definition ∂ ′ := d ′ P − d ′ for all polyhedral currents in the following.Pick any affine linear map f : R n → R dim τ +1 such that f | σ is injective for every τ ⊂ σ ∈ T r .Let C = τ ⊂σ σ be the polyhedral set formed by all σ ∈ T r containing τ .Denote by Z its boundary in the topological space σ∈T r σ.The current S = τ ⊂σ α σ ∧ [σ, µ σ ] has support contained in C and is a δ-form away from Z. Since f | C has finite fibers, Supp S is relatively compact over R m , so the pushforward f * S is defined.It is a δ-form away from f (Z).Moreover f * (d f * S is given by a piecewise smooth form away from f (Z) by Lem.3.7.This makes the determination of γ very simple: Picking the normal vectors in the above (3.9)as n := n ρ1,f (τ ) = −n ρ2,f (τ ) eliminates the first sum in (3.9) and shows (3) For the third and final formula, choose coordinate functions x 1 , . . ., x n−r−1 : R n → R that restrict to a basis of M τ .For each σ containing τ , choose a non-constant affine linear function z σ : σ → R such that z σ | τ is constant.(For example, one may choose an affine linear z : R n → R such that z| τ is constant but z σ = z| σ non-constant for every σ ⊃ τ .)Then every α σ can be uniquely expressed as Our claim is that Note that already the individual summands are independent of the chosen z σ .
Proof of the claim.In light of (3.3), we need to show that the following identity holds for all smooth forms η ∈ A n−p−r−1,n−q−r (R n ) of complementary degree: Our task is thus to show Pick coordinate functions y 1 , . . ., y r+1 : R n → R that extend x 1 , . . ., x n−r−1 to a basis and that are constant along τ .Then It is thus left to show (3.13) for forms η = d ′′ y i ∧ η.We obtain that The last expression vanishes by the balancing condition (3.2).

Main result
The definition of the ∧-product of δ-forms is based on two specific constructions.The first is the product of piecewise smooth and δ-forms from Ex. 3.8: By Lem.3.7, every δ-form of tridegree (p, q, 0) is of the form α ∧ [R n , µ std ] for a (unique) piecewise smooth (p, q)-form α.We write α by abuse of notation and define The second construction is the exterior product of currents, cf.[4, §I.2], defined as follows.Given homogeneous currents The exterior product preserves polyhedral currents which follows from the identity Relation (4.2) then implies that the exterior product of δ-forms is a δ-form again.Moreover, one sees that if T i is of polyhedral tridegree (p i , q i , r i ), then T 1 ⊠T 2 has tridegree (p 1 +p 2 , q 1 +q 2 , r 1 +r 2 ).Separating (4.2) by tridegree provides We simply write T 1 × T 2 instead of T 1 ⊠ T 2 for δ-forms T 1 and T 2 .
In the following, ∆ = (id, id) * [R n , µ std ] ∈ B 0,0,n (R n × R n ) denotes the diagonal viewed as δ-form.This product has the following additional properties.
(1) It is graded commutative and trihomogeneous in the sense B p,q,r ∧ B s,t,u ⊆ B p+s,q+t,r+u .In particular, it satisfies the Leibniz rule with respect to the operators ∂ ′ , d ′ P , ∂ ′′ and d ′′ P .
(2) It commutes with pull-back: Given a surjective affine linear map f : R n → R m and S, T ∈ (3) It satisfies the projection formula: Given a surjective affine linear map f : (4.6) (4) It coincides with the tropical intersection products from [1,5,9] on r B 0,0,r whenever they are defined.
The idea of characterizing and constructing the tropical intersection product through divisor intersections and restriction to the diagonal is due to Allermann-Rau [1].
Proof of the uniqueness assertion.If a ∧-product exists as claimed, the Leibniz rule implies for piecewise smooth α that In case of a piecewise linear function ϕ, the δ-forms d ′ ϕ resp.d ′′ ϕ agree with d ′ P ϕ resp.d ′′ P ϕ and are again piecewise smooth, because the contractions in (3.9) vanish for degree reasons.(This applies more generally to piecewise smooth functions.)It follows that is uniquely determined by the Leibniz rule and the piecewise smooth case.Denote by x 1 , . . ., x n and y 1 , . . ., y n the coordinate functions on R n × R n and let ϕ i := max{x i , y i }.Then, by [1, Rmk.9.2], the diagonal ∆ is the product where the right hand side is a successive application of (4.8).Again by (4.8) as well as the associativity of the ∧-product, ∆ ∧ (S × T ) is now uniquely determined.Hence S ∧ T = p 1, * (∆ ∧ (S × T )) is uniquely characterized by the stated conditions.
The existence statement will be shown in the next section.Here, we give an application of Thm.4.1 to the definition of a pull-back for all affine linear maps, not just surjective ones.It is specific to δ-forms, meaning it does not extend to polyhedral currents.Its construction is well-known for tropical cycles, cf.[9, Rmk.1.4 (v)] for example.
Proposition/Definition 4.2.Let f : R n → R m be an affine linear map and S ∈ B(R m ) a δform.There is a unique δ-form f * (S) ∈ B(R n ), called the pull-back of S along f , that satisfies the projection formula This pull-back is functorial in f and commutes with ∧-products as well as all the six differential operators.
Proof.Identity (4.9) determines f * S uniquely because it determines all its values on test forms η by Just from this uniqueness, one may deduce all further properties.For example, for d ∈ {d ′ , d ′′ } and for all δ-forms T , For commutativity with ∧-products, we compute We omit the verification of the remaining properties which are shown similarly.
To show existence of f * , we consider the graph We claim that the following definition satisfies (4.9): The next succession of identities verifies that claim.The first four equalities come either by definition or from the projection formula (4.6).The last equality will be explained below.The last equality comes from the identity p 2, * (p * 1 T ∧ Γ f ) = f * T which may be seen as follows.The form p * 1 T ∧ Γ f has support contained in Supp Γ f and has the property It is then merely left to note that p 2 • (id, f ) = f and the proof is complete.

Example 4.3. Every affine linear map
Especially interesting here is the property of h * to commute with ∧-products, cf.Prop.4.2.It specializes to where ∧ L denotes the wedge product on L.

Existence of the ∧-product
This section proves the existence of the ∧-product.We begin with some Leibnize rule properties of the product with piecewise smooth forms in (4.1).
(1) For every homogeneous piecewise smooth form α and polyhedral derivative d P ∈ {d ′ P , d ′′ P }, (2) For every homogeneous piecewise smooth form α ∈ B p,0,0 , (3) Analogously, for every homogeneous piecewise smooth form α ∈ B 0,q,0 , Proof.Identity (1) may be checked polyhedron by polyhedron and, in this way, reduces to the Leibniz rule for smooth forms.Identities ( 2) and ( 3) follow from the observation that the contractions in (3.9) are linear (up to the sign (−1) deg α ) with respect to multiplication by piecewise smooth functions in the stated degrees.
By a divisor we mean a d ′ -closed and d ′′ -closed δ-form of tridegree (0, 0, 1).These are the tropical cycles with constant coefficients of codimension 1 in classical terminology.Proof.This is entirely due to Lagerberg, cf.[11,Prop. 5.3], we merely give the straightforward reduction to his results.Let U 1 ⊂ U 2 ⊂ . . .be a covering of R n by convex relatively compact opens.By definition, D is a locally finite sum of currents m • [σ, µ] with m ∈ R.So for each i ≥ 1, there is a finite linear combination H i of weighted hyperplanes such that (D + H i )| Ui is positive in the sense that all its coefficients are ≥ 0. By [11, Prop.2.4 and Prop.2.6], there is then a convex function The lemma is easily seen to hold for hyperplanes and hence the H i , so we obtain for each i the existence of a convex function ϕ i with D| Ui = d ′ d ′′ ϕ i .Then ϕ i is necessarily piecewise linear, cf.[11,Proof of Prop. 5.3].A piecewise linear function ϕ is affine linear if and only if d ′ d ′′ ϕ = 0, so the ϕ i are determined up to addition of affine linear functions.They may then be chosen compatibly, i.e. such that they satisfy ϕ i+1 | Ui = ϕ i , proving the lemma.
For affine linear ϕ and every current T we have by (2.9) the relation 12) The definition does not depend on the choice of ϕ by (4.11).The resulting D • T is again a δ-form.Lemma 4.8.Let D, T and ϕ be as above.The following two identities hold: Proof.Part (3) of Lem.4.4 shows that Substituting this in (4.12) immediately leads to the first equality of (4.13).
Part (1) of Lem.4.4 together with the observation d ′ P d ′′ ϕ = 0 implies that Substituting this in (4.13) gives the second equality.
We remark that identity (4.13) collapses to the definition of the corner locus [9, Def.1.10] if T is a tropical cycle.Also, if ϕ is affine linear, then d ′′ ϕ is a smooth form and sign-commutes with ∂ ′ by Lem.4.4.Then (4.13) gives d ′ d ′′ ϕ • T = 0 as expected.The identity also shows that if T is of tridegree (p, q, r), then D • T is of tridegree (p, q, r + 1).Its most important consequence for us, however, is the following simple description of D • T .Lemma 4.9.Let ϕ be a piecewise linear function and T a δ-form of tridegree (p, q, r).Let further T be a weighted polyhedral complex subordinate to both ϕ and T , say with Assume first that (ϕ − ϕ τ )| σ is non-constant for every σ ∈ T r containing τ .Then we can put z σ = (ϕ − ϕ τ )| σ to obtain (4.14) from a literal application of (3.11).
The general case follows since the right hand side of (4.14) is a priori independent of the choice ϕ τ by the balancing condition (3.2).

Let x and y denote the coordinates on the first two factors
It becomes a weighted complex by endowing σ ≥ and σ ≤ with weight µ std ∧µ σ and g(σ) with g(µ σ ).
Recall that Then the following equalities hold, as will be explained below.
The first equality is the definition of the left hand side combined with the identity f The second follows from the projection formula for divisor intersection, Lem.4.11, applied to ∆ m = (f, f ) * δ m , where δ m ⊂ R m ×R m denotes the diagonal.The third equality is the observation for any divisor C and δ-forms X, Y , applied successively to the divisor intersection ∆ c .The map g in the next line is the partial diagonal and the identification ∆ c • (R c × T ) = g * T is Lem.4.12.The final equality then is the observation (c) The next claim is that the ∧-product is associative, S ∧ (T ∧ U ) = (S ∧ T ) ∧ U .Indeed, applying the projection formula (b) repeatedly, one obtains where the intersection takes place on (R n ) 4 and where 4 .In exactly the same way, The two expressions are seen to be equal by switching the middle factors as in Step (a).
(d) Next, we claim that α ∧ T = αT for every piecewise smooth α.This follows from Lem. 4.12 and the fact that multiplication by piecewise smooth forms commutes with divisor intersection.The latter is immediate from Lem. 4.9.
(f) We turn to the Leibniz rule.Let C be a divisor and T a δ-form.Our first step is to prove the identity for a piecewise linear function ϕ as in Lem.4.5 and using (4.13) twice, we have This finishes the proof of (4.20).Successive application of the divisor case now yields The Leibniz rule (4.2) for exterior products, coupled with Def.4.13, completes the proof of the Leibniz rule for d in general.Separating by tridegree provides the Leibniz rules for the other differential operators.(h) Finally, the tropical intersection products of Allermann-Rau [1], its extension to smoothly weighted rational polyhedra in [9,Rmk. 1.4], and the intersection product of Esterov [5] for polynomially weighted (possibly non-rational) polyhedra can all be expressed in terms of divisor intersection and restriction to the diagonal.In these two specific cases, they coincide with our definition.So any two of the mentioned products coincide whenever both are defined.

Fan Displacement Rule
Two linear subspaces N 1 , N 2 ⊆ R n are said to intersect transversally if N 1 + N 2 = R n .(Equivalently, their intersection is transversal if codim(N 1 ∩ N 2 ) = codim(N 1 ) + codim(N 2 ).)In the transversal case, there is an exact sequence Given weights µ 1 and µ 2 for N 1 and N 2 , respectively, we denote by µ 1 ∩ µ 2 the weight on N 1 ∩ N 2 that satisfies (µ 1 ∩ µ 2 ) ∧ µ std = µ 1 ∧ µ 2 in the sense of (2.12).The next lemma is easily checked.Lemma 4.17.Let [N 1 , µ 1 ], [N 2 , µ 2 ] ⊆ R n be weighted linear subspaces, viewed as δ-forms.Assume that their intersection is transverse.Then Let T 1 and T 2 be polyhedral complexes on R n which are pure of codimension r 1 and r 2 , respectively.By this we mean that T i agrees with the set of faces of all σ i ∈ T ri i .Then T 1 and T 2 are said to intersect transversally if, for all pairs (σ 1 , σ 2 ) ∈ T r1 × T r2 , the intersection σ 1 ∩ σ 2 is either empty or of codimension r 1 + r 2 and not contained in the union of boundaries ∂σ 1 ∪ ∂σ 2 .Note that then N σ1 and N σ2 intersect transversally whenever σ 1 ∩ σ 2 = ∅.
Construction 4.20.Let v be a generic vector for two polyhedral complexes T 1 and T 2 that are pure of codimensions r 1 and r 2 , respectively.Let The sum here is over all (σ 1 , σ 2 ) such that σ 1 ∩ (εv + σ 2 ) = ∅ for all sufficiently small ε.Note that v need not be generic for subdivisions of T 1 and T 2 anymore, which is why the definition depends on their choice.
Proposition 4.21.Let T 1 and T 2 be δ-forms with subordinate polyhedral complexes T 1 and T 2 as above.Assume v is generic for T 1 and T 2 .Then the v-displacement product (with respect to the T i ) computes the ∧-product,

.27)
In particular, the v-displacement product is independent of the choices T 1 , T 2 and v.
Proof.Both sides of (4.27) are computed locally, so we may assume T 1 and T 2 to be finite by a partition of unity argument.Lem.4.18 then shows that for all sufficiently small ε > 0, The intersection σ 1 ∩ (εv + σ 2 ) being non-empty and transverse for all sufficiently small ε implies that σ 1 ∩ σ 2 is non-empty and of codimension r 1 + r 2 .Moreover in this case, in the weak sense.It follows that T 1 ∧ (εv + T 2 ) → T 1 • v T 2 in the weak sense.Prop.4.19 on the other hand shows that this limit equals T 1 ∧ T 2 , proving the proposition.

Theorem 4 . 1 .
There is a unique way to define an associative product ∧ : B × B → B that satisfies the Leibniz rules with respect to d ′ and d ′′ , extends definition (4.1), and can be computed by restriction to the diagonal, meaning S ∧ T = p 1, * (∆ ∧ (S × T )).(4.5)

Lemma 4 . 5 .
Let D be a divisor on R n .Then there exists a piecewise linear function ϕ such that D = d ′ d ′′ ϕ.It is unique up to addition of affine linear functions.

. 11 ) 4 . 6 .
Definition Let D be a divisor and T a δ-form.Choose a piecewise linear function ϕ with D = d ′ d ′′ ϕ as in Lem.4.5 and define

Lemma 4 . 11 .
Let f : R n → R m be a surjective affine linear map, T a δ-form on R n with compact support over R m and D a divisor on R m .Then the projection formula holds,D • f * T = f * (f * D • T ).Proof.Write D = d ′ d ′′ ϕ for a piecewise linear function ϕ as in Lem.4.5.Push-forward commutes with both d ′ and d ′′ while multiplication with the piecewise smooth forms ϕ, d ′ ϕ and d ′′ ϕ on R m in the sense of Ex. 3.8 obviously satisfies the projection formula.The claim now follows directly from Def. 4.6.

( g )
The identity f * (S ∧ T ) = f * S ∧ f * T only uses the fact p * 1 D • (S × T ) = (D • S) × T for divisor intersection.Namely assume f : R m × R c → R m to be the projection and write p 12 : R m × R c × R m × R c → R m × R c for the projection to the first two factors.Then, in the terminology of Step (b),f * S ∧ f * T = p 12, * (∆ m • ∆ c • (S × R c × T × R c )) = p 12, * (∆ m • (S × T )) × ∆ c = (S ∧ T ) × R c = f * (S ∧ T ).