A general framework for tropical differential equations

We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the norms of the coefficients. This gives a significant refinement of Grigoriev's framework for tropical differential equations. We then prove a differential analogue of Payne's inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.


INTRODUCTION
Algebraic ODEs are systems of differential equations formed from polynomial expressions in an indeterminate function f and its derivatives.The algebraic theory was first developed by Ritt [24] and collaborators.Many important classes of models from the natural sciences, such as chemical reaction networks, are algebraic ODEs, and in pure mathematics algebraic ODEs appear in many parts of geometry, including periods and monodromy.Understanding their solutions and singularities has many important consequences in pure and applied mathematics.
In this paper we pursue the further development of the tropical mathematics tool set for studying differential equations.Tropical geometry is a field at the interface between combinatorics, computational algebra/geometry, and algebraic geometry.One of the foci of tropical geometry is the study of non-archimedean amoebae of affine varieties over non-archimedean normed fields, which can be viewed as combinatorial shadows of varieties.Tropical geometry has provided important computational and theoretical tools for algebraic geometry, and we hope to open the door to similar applications in differential algebra.
In [11] Grigoriev first introduced a theory of tropical differential equations and defined a framework for tropicalizing algebraic ODEs over a ring of formal power series R [[t]].In this framework, one tropicalizes a differential equation by recording the leading power of t in each coefficient, and one tropicalizes a power series solution simply recording the powers of t that are present.
Solutions to a differential equation tropicalize to solutions to its tropicalization, and Grigoriev asked if all solutions to the tropicalization of an equation arise as tropicalizations of classical solutions; i.e., is the map from classical solutions to tropical solutions surjective?This is the differential equation analogue of the Fundamental Theorem of Tropical Geometry [22,Theorem 3.2.3],and this question was answered positively by Aroca et al. in [1] (assuming R is an uncountable algebraically closed field of characteristic 0).These ideas have also been extended to the case of algebraic partial differential equations in [6].
Paralleling the role of Gr öbner theory in defining tropical varieties in the non-differential setting, [7] and [12] define initial forms and develop a Gr öbnertheoretic approach to Grigoriev's tropical differential equations.A similar approach is also presented in [4], which also gives an illuminating account of tropical ordinary and partial differential equations (in part based on a preliminary report of the algebraic perspective presented here).
A limitation present in all of the above work is that the tropicalization construction studied there records only the powers of t present in a power series solution; it does not record any information about the norms of the coefficients.Thus any information about convergence of power series solutions is lost when using Grigoriev's tropicalization.In the theory of p-adic differential equations, one of the central problems is to understand the radii of convergence of formal power series solutions, which are controlled by the norms of the coefficients.
1.1.Results.The main purpose of this work is to build a refinement of Grigoriev's framework that records and incorporate the norms of the coefficients in a power series solution so that convergence information is encoded in tropical solutions.This requires developing a theory of differentials on idempotent semirings in which the usual Leibniz rule is weakened to a tropical Leibniz rule, and this development includes constructing free tropical differential algebras (a tropical analogue of Ritt algebras).
We give a brief explanation of our framework here.A tropical pair S = (S 1 → S 0 ) is a tropical differential semiring S 1 and a homomorphism to a semiring S 0 .The coefficients of tropical differential equations live in S 0 .Solutions live in S 1 (where they can be differentiated), but the condition that tests if something is a solution takes place in S 0 .We think of S 0 as recording the leading behavior of elements of S 1 .The primary example of a tropical pair has S 1 = T[[t]] (the semiring of formal power series with tropical real number coefficients), S 0 = R 2 lex ∪ {∞} is a rank 2 version of the tropical semiring, and the map S 1 → S 0 sends at n + • • • to (n, a).We now state out main results informally.
Theorem A. We construct a category of S-algebras, and to a set E of tropical differential equations over S we associate an object of this category such that morphisms to an Salgebra T are in natural bijection with solutions to E with values in T.
A system of algebraic differential equations over a field k is represented in coordinate-free form by a differential k-algebra A. To tropicalize A, we need two pieces of additional data: (1) A non-archimedean norm on k taking values in an idempotent semiring S 0 , and a differential enhancement of the norm, which is a lifting to a map A → S 1 that commutes with the differential.(These notions are defined in Sections 2.3 and 4.7.)(2) A system of generators x i ∈ A so that A is presented as a quotient of a Ritt algebra k{x 1 , . . ., x n } ։ A. Any differential algebra A admits a universal presentation k{x a | a ∈ A} → A. Tropicalizing this presentation, we find:

Theorem B. The tropicalization of A with respect to its universal presentation is the colimit of its tropicalizations with respect to finite presentations.
Finally, we provide evidence for the appropriateness of our definitions and framework by proving a differential analogue of Payne's inverse limit theorem [23].Recall that, given an algebra A over a non-archimedean field k, the underlying set of the Berkovich analytification of Spec A is the set of all multiplicative seminorms on A that are compatible with the norm on k.Now suppose that k is a differential ring, the norm v on k has a differential enhancement v taking values in a pair S, and A is a differential algebra over k.In this setting, given an S-algebra T = (T 1 → T 0 ), we can consider the set of all pairs (w, w) where w : A → T 0 is a multiplicative seminorm on A compatible with v and w : A → T 1 is a differential enhancement of w compatible with v.We call this the T-valued differential Berkovich space of A, denoted Berk T (A).
Acknowledgements.We thank Alex Fink and Zeinab Toghani for many useful discussions and for organizing an influential workshop at QMUL in December 2019 in which the tropical differential equations community came together and shared many ideas.We also thank Andrea Pulita for his encouragement to do this.S.M. gratefully acknowledges the support of a Swansea University Strategic Partnerships Scholarship and funding from Université Grenoble Alpes IDEX.
2.1.Idempotent semirings.An idempotent semiring is a semiring (S, ⊕, ⊗) in which addition is an idempotent operation: a ⊕ a = a.An idempotent semiring carries a canonical partial order defined by a ≤ b if a ⊕ b = b.The additive unit 0 S is the unique minimal element.In a semiring, we will often write the product a ⊗ b simply as ab.
(2) Let T n denote (R >0 ) n ∪ {(0, . . ., 0)}.We define a ⊕ b = max(a, b), where the maximum is taken with respect to the lexicographic ordering on the positive orthant, and we define a ⊗ b to be component-wise multiplication.(3) The boolean semiring B is the sub-semiring {0, 1} ⊂ T. Note that B is the initial object in semirings.

Congruences and bend relations.
A quotient of a ring R is defined by an equivalence relation on R such that the ring structure descends to the set of equivalence classes.Such equivalence relations are of course in bijection with ideals via the correspondence This correspondence does not hold for semirings in general, and so we must work with the equivalence relations themselves when defining quotients.
Recall that a congruence on a semiring S is an equivalence relation K ⊂ S × S that is also a subsemiring.If K is a congruence on S, then the semiring structure on S descends to a well-defined semiring structure on S/K.Moreover, if f : S ։ S ′ is a surjective homomorphism of semirings then its kernel congruence ker Given a set of binary relations X ⊂ S × S, the congruence it generates can be described concretely.First take the subsemiring of S × S generated by X, and then take the transitive and symmetric closure of this.See [9,Lemma 2.4.5].
We now come to a class of congruences on idempotent semirings that are essential in tropical geometry.Given an expression a 1 ⊕ a 2 ⊕ • • • ⊕ a n in an idempotent semiring S, the bend relations of this expression, written B(a is the congruence on S generated by the relations The motivation for the bend relations stems from the following fact.Recall that the tropical hypersurface of a tropical polynomial f ∈ T[x 1 , . . ., x n ] can be described away from the boundary of T n as the locus where the graph of f is non-linear (with respect to the R ∪ {∞} parametrization of T).Proposition 2.2.1 (Prop.5.1.6 of [9]).Given a tropical polynomial f ∈ T[x 1 , . . ., x n ], the tropical hypersurface of f is precisely the set of homomorphisms T[x 1 , . . ., x n ]/B( f ) → T.

Non-archimedean seminorms.
A non-archimedean multiplicative seminorm on a ring R is a map v from R to the tropical semiring T = (R ≥0 , ⊕ = min, ×) that is a homomorphism of multiplicative monoids, has v(0) = 0, and satisfies the ultrametric triangle inequality, The map −log : R ≥0 → R ∪ {∞} identifies non-archimedean seminorms (always assumed to be multiplicative) with valuations, and so we will use the terms interchangeably as we adopt the following generalization.Definition 2.3.1.A non-archimedean seminorm on a ring R is an idempotent semiring T and a map v : R → T satisfying (1) Remark 2.3.2.Condition (4) generalizes the ultrametric triangle inequality, as it says that v(a + b) ≤ v(a) ⊕ v(b) in the canonical partial order on T. This definition thus becomes equivalent to the usual definition of a Krull valuation when the partial order is a total order, such as when T = T n .This condition can also be written more symmetrically as For use later on, we record the following simple observation.A rank 1 valuation v : R → T can be extended to a rank 2 valuation on the the ring R{{t}} of Puiseux series (or the subrings of formal Laurent series or polynomials) by the formula (2.3.1) 2.4.Tropical differential semirings.Recall that a differential ring is a ring R equipped with an additive map d : R → R that satisfies the Leibniz relations d(ab) = (da)b + a(db).This relation for 2-fold products easily implies that d also satisfies an analogous relation for n-fold products.We will call these relations the strict Leibniz relations, and a map d satisfying them will be called a strict differential.
In this work we propose that differentials on idempotent semirings should be required to satisfy a somewhat weaker condition than the strict Leibniz relations.Given an idempotent semiring S, an additive map d : S → S is said to be a tropical differential if it satisfies the tropical Leibniz relations: for any pairs of elements x, y ∈ S the bend relations of the expression hold.Note that we can view the tropical Leibniz relations as the tropicalization of the strict Leibniz relations.
Indeed, this satisfies the tropical Leibniz relations since

on the right satisfies the bend relations (the argument is essentially the same for k-fold products with k > 2).
Note that v could be either a p-adic norm, or a degenerate p-adic norm where v(n) = 0 if p divides n, and 1 otherwise.
Remark 2.4.4.While B is the initial objects in the category of idempotent semirings, tropical differential semirings do not admit an initial object because the tropical Leibniz rule does not determine d(1).
2.5.Differential congruences.Let (S, d) be a semiring equipped with an additive map d : When K is a differential congruence, the map d descends to an additive map d : S/K → S/K, and if d is a tropical differential then d is as well.
Proposition 2.5.1.If {I λ ⊂ S × S} is a set of differential congruences, then the congruence generated by them is a differential congruence.
Proof.Let K denote the congruence generated by the I λ .It is the transitive and symmetric closure of the subsemiring K 0 generated by the I λ .
We will first show that d(K 0 ) ⊂ K. Suppose that (a 1 , a 2 ) is a relation in some For the product, we proceed as follows.In ) is indeed contained in the transitive closure K. Since any element of K 0 is produced by a finite sequence of sums and products of elements in the I λ , it follows that d(K 0 ) ⊂ K, as desired.Now, any relation in K can be decomposed as a finite transitive chain of relations in K 0 .Thus is follows that d(K) ⊂ K.

DIFFERENTIAL POLYNOMIALS
The objective of this section is to construct a variant of the Ritt algebra of differential polynomials R{x 1 , . . ., x n }, where the coefficient ring R is replaced by a tropical differential semiring S.

Classical Ritt algebras and their universal property.
Recall that, when R is a differential ring, the Ritt algebra of differential polynomials R{x 1 , . . ., x n } is the commutative R-algebra freely generated by the variables x i and their formal derivatives x and extends as an additive derivation to arbitrary elements in R{x 1 , . . ., x n }.
Ritt algebras are characterized by a universal property: a homomorphism of differential rings ϕ : R{x 1 , . . ., x n } → R ′ is uniquely determined by the images of the generators ϕ(x i ) ∈ R ′ , and there are no restrictions on these elements.It is this universal property that we wish to extend to the idempotent setting.

3.2.
The failure of the naive definition of tropical Ritt algebra.While one can trivially replace the coefficient ring R with a differential idempotent semiring S in the above Ritt algebra construction, we shall now see that it impossible to endow this with a differential (either strict or tropical) for which the analogous universal property holds.In fact, we will show that there is no longer a unique choice of differential, and for any choice of differential the universal property fails.
Suppose that S is a differential idempotent semiring with tropical differential d S .The naive definition of differential polynomials over S suggested in the preceding paragraph will nevertheless be useful later on, so we give it a name: Definition 3.2.1.Given a tropical differential semiring S, the algebra of basic differential polynomials over S, denoted S{x 1 , . . . ,x n } basic is the polynomial S-algebra on variables x i and their formal derivatives x (j) i .
Let us now attempt to extend the differential of S to this algebra.Obviously we would like to send x , and we would like the map to be additive.The difficulty is in choosing how to extend it to arbitrary products.In contrast to the case of coefficients in a differential ring, the tropical Leibniz relations allow more freedom in extending a partially-defined differential to all products; there is not a uniquely determined extension of d S to a map d on all of S{x 1 , . . ., x n } basic satisfying the tropical Leibniz relations.In fact, S{x 1 , . . ., x n } basic admits many distinct differentials.Example 3.2.2.Suppose w : N → S is a non-archimedean seminorm.Then we can define a differential d w on S{x 1 , . . ., x n } basic by the following rule.First, for a pure power (x Then, we extend this to monomials c(x

It is straightforward to check that this map d w does indeed satisfy the tropical Leibniz relations. One can generalize this example by choosing a distinct non-archimedean seminorm
w ij for each generator x i , defining the differential on pure powers by the rule d((x , and then extending to arbitrary monomials using the strict Leibniz rule.
The above example shows that there is at least one differential on S{x 1 , . . ., x n } basic for each n-tuple of non-archimedean seminorms Q → S.

Proposition 3.2.3.
There is no tropical differential on S{x 1 , . . ., x n } basic that extends the tropical differential on S and makes this the free object on n generators.
Proof.We use proof by contradiction.Suppose d is such a differential, so for any non-archimedean seminorm w : N → S the identity map must be a morphism of differential idempotent semirings This implies that d = d w on pure powers.But this is a contradiction since d w = d w ′ if w and w ′ are distinct seminorms.
We will show below that the idempotent semiring S{x 1 , . . ., x n } basic can be enlarged to a tropical differential semiring S{x 1 , . . ., x n } enjoying the universal property that justifies calling it the tropical Ritt algebra.

Differential polynomials over a ring.
As a warm-up to constructing tropical Ritt algebras, we first give an alternative construction of the classical Ritt algebras in terms of trees.
To avoid ambiguity, let us be precise about our graph theory definitions and conventions.Definition 3.3.1.A forest is a finite set V (the vertices) together with a parent map P : V → V such for n large enough P n sends each vertex to a fixed point of P. The fixed points of P are the roots.A tree is a forest with a single root.The valence of a vertex v ∈ V is the cardinality of P −1 (v) {v}, and the vertices of valence 0 are the leaves.
Consider the set of isomorphism classes of forests with leaves labelled by elements of R ∪ {x 1 , . . ., x n }.This set becomes an abelian monoid under the operation of disjoint union.Now let Forest(R, n) denote the quotient monoid obtained by imposing the following relations: (1) A tree with a leaf labelled by 0 ∈ R is identified with the empty forest.
(2) Any leaf labelled by 1 can be deleted.(5) Given elements r 1 , r 2 ∈ R and a tree t, we may form trees r 1 • t, r 2 • t, and (r 1 + r 2 ) • t by grafting a leaf with label r 1 , r 2 , and (r 1 + r 2 ), respectively, at the root.We then identify the pair of trees (r We think of the set of leaves with a given parent as representing the monomial formed by multiplying their labels, and we think of internal edges not incident at the root as representing the differential d.Thus a tree represents a differential monomial, i.e., an expression formed from the elements of R and the variable x 1 , . . ., x n by taking products and applying the differential d. Proof.The addition operation is disjoint union (the original monoid operation).The product of two trees is defined by gluing their roots together, and we extend this operation to products of forests by the distributive rule.The elements r ∈ R sit inside Forest(R, n) as the trees consisting of just a root and a single leaf with label r.It is straightforward to verify that this is an R-algebra.
There is a map d : Forest(R, n) → Forest(R, n) that inserts an edge at each root in a forest: d → This map is clearly not a derivation, so let and let L denote the differential ideal generated by L, i.e., the smallest ideal in Forest(R, n) that is closed under applying d.By construction, the differential d descends to a derivation on the quotient.

Proposition 3.3.4. There is a natural isomorphism of differential R-algebras,
Proof.It is straightforward to check that Forest(R, n)/ L has exactly the same universal property as the Ritt algebra.
3.4.Differential polynomials over a semiring.We now give a variation on the above construction of the Ritt algebra via trees, where we replace R with a differential idempotent semiring S, and we replace the classical Leibniz relations with the tropical Leibniz relations.
First consider the S-algebra Forest(S, n) defined exactly as above, and then consider the subset Let L trop denote the smallest ideal containing L trop and closed under applying d.Definition 3.4.1.Given a differential idempotent semiring S, we define the tropical Ritt algebra S{x 1 , . . . , where B L trop is the congruence of bend relations generated by L trop .
The tropical Ritt algebra enjoys a universal property in the category of differential idempotent semirings that is entirely analogous to the universal property of the classical Ritt algebra in the category of differential rings.Proposition 3.4.2.Given a differential S-algebra S ′ , there is a bijection implemented by sending a homomorphism ϕ to the n-tuple (ϕ(x 1 ), . . ., ϕ(x n )).The algebra S{x 1 , . . ., x n } basic of basic differential polynomials that was introduced earlier sits inside S{x 1 , . . ., x n } as the set of forests where only the root vertices have valence larger than 1; as the notation suggests.Obviously the basic subalgebra is not closed under taking differentials.

ALGEBRAIC STRUCTURES FOR TROPICAL DIFFERENTIAL EQUATIONS
In the classical world, a differential equation over a differential ring R is an element f ∈ R{x 1 , . . ., x n }, and a solution to f in an R-algebra A is an element p ∈ A n such that f (p) = 0. Equivalently, p is a solution if the corresponding homomorphism p : R{x 1 , . . ., x n } → A factors through the quotient R{x 1 , . . . ,x n }/( f ).
In the tropical world, we have introduced differential idempotent semirings, but these objects on their own are not sufficient to describe solutions to tropical differential equations.A tropical differential equation over a differential idempotent semiring S is an element f ∈ S{x 1 , . . ., x n } (where this is the tropical Ritt algebra defined above).Solutions to this differential equation will live in S n , but asking that f vanish or tropically vanish at p ∈ S n turns out to be too restrictive.Following the idea of Grigoriev's framework, p should be considered a solution to f if f tropically vanishes at p to leading order (rather than to all orders).This suggests that we must equip our differential idempotent semirings with something like a non-archimedean seminorm that provides a way of measuring the leading order of elements.To this end, we will now define and study the category of tropical pairs.4.1.The category of tropical pairs.A tropical pair S consists of a tropical differential semiring S 1 , an idempotent semiring S 0 , and a homomorphism of idempotent semirings π : S 1 → S 0 .
Remark 4.1.1.We think of S 1 as a space of functions, and we think of S 0 as a space of leading exponents of the series expansions of these functions.The map π, like the usual norm on Puiseux series, sends a function to its leading exponent.
In category theoretic terms, if F : DiffSemirings → Semirings is the forgetful functor from differential idempotent semirings to idempotent semirings, then the category of pairs is the simply the comma category (F ↓ Semirings).Explicitly, a morphism of pairs ϕ from (S in which the upper horizontal arrow ϕ 1 is a morphism of differential idempotent semirings.Given a pair S, the category of S-algebras is the category of pairs T under S.
A pair (S 1 π → S 0 ) is said to be reduced if S 1 admits no nontrivial quotient differential idempotent semiring over S 0 ; i.e., it is reduced if there is no nontrivial differential congruence contained in the congruence ker(π).( This is a pair, and a modification of the argument above shows that it is also reduced. We let Pairs red denote the full subcategory of reduced pairs.We will show below in Section 4.3 that Pairs red is a reflective subcategory, and so any pair S = (S 1 → S 0 ) has a functorial reduction S red = (S red 1 → S 0 ).Finally we are ready to define the category that will describe tropical differential equations and their solutions.Definition 4.1.3.Given a reduced pair S, an S-algebra is a reduced pair under S, and we let S-Alg denote the category of S-algebras.
An important example of an S-algebras comes from the tropical Ritt algebra.Given a pair S = (S 1 → S 0 ), we first define an idempotent semiring (S 0 |S 1 ){x 1 , . . ., x n } by taking the pushout: This pushout can be described explicitly as the algebra of trees S 1 {x 1 , . . ., x n } modulo the congruence generated by the relations that identify a, b ∈ S 1 ⊂ S 1 {x 1 , . . ., x n } if they have the same image in S 0 ; i.e. leaves incident at the root have labels in S 0 rather than S 1 .Note that (S 0 |S 1 ){x 1 , . . ., x n } contains the polynomial S 0 -algebra S 0 {x 1 , . . ., x n } basic .The right vertical arrow in the above diagram gives an S-algebra that we will denote by S{x 1 , . . ., x n }; these pairs will play the role of tropical Ritt algebras in the category of S-algebras. 1 .The bijection is implemented by sending a morphism ϕ = (ϕ 1 , ϕ 0 ) to (ϕ 1 (x 1 ), . . .ϕ 1 (x n )).

Proof. Given y ∈ Y n
1 , it follows from the universal property of the tropical Ritt algebra (Prop.3.4.2) that there is a unique morphism of S 1 -algebras ϕ 1 : S 1 {x 1 , . . ., x n } → Y 1 sending x i to y i .By the universal property of pushouts, this induces an arrow ϕ 0 : (S 0 |S 1 ){x 1 , . . ., x n } → Y 0 such that (ϕ 1 , ϕ 0 ) is a morphism of S-algebras, and this is unique since ϕ 1 is unique.

Tropical differential equations and their solutions.
We start with a reduced pair S = (S 1 π → S 0 ).A tropical differential equation is simply a differential polynomial f ∈ S 0 {x 1 , . . ., x n } basic .Let us write f = ∑ α f α x α , where x α runs over the differential monomials in f .If x α has any factors of the form d n x i for n > 0 then it does not make sense to evaluate x α at an element c ∈ S n 0 because S 0 is not a differential semiring.However, we can evaluate x α at an element C ∈ S n 1 and then push down to S 0 via π.Thus we can evaluate f ∈ S 0 {x 1 , . . ., tropically vanishes.

When the pair S 1 → S 0 is B[[t]]
π → T, the above definition recovers Grigoriev's framework.A subset W ⊂ N corresponds to the boolean formal power series i∈W t i , and Grigoriev's map Val W (j) is precisely π(d j W).
The maximum occurs only in the middle term, so there is no solution with α = 0.
The second and third terms are equal and maximal, so this is a solution for any nonzero value of β.Proof.Suppose T = (T 1 ψ → T 0 ) is a reduced pair and is a morphism of pairs.There are inclusions The map ϕ 1 sends ker ϕ 0 • π into ker ψ, and the image of a differential congruence by a homomorphism of tropical differential semirings is again a differential congruence, so ϕ 1 must send R(π) to a differential congruence contained in ker ψ.
Since T is reduced, the only such differential congruence on T 1 is the diagonal, and so ϕ 1 factors uniquely through the quotient map S 1 → S 1 /R(π).
We now show that the above reduction construction exhibits Pairs red as a reflective subcategory of Pairs.Proof.Suppose f : S → T is a morphism of pairs and consider the composition S → T → T red .By Proposition 4.3.1,there is a unique factorization S → S red → T red , and hence we obtain a morphism R( f ) : S red → T red .It is straightforward the check that this respects compositions: R( f It is a straightforward verification that the quotient map S → S red defines a natural transformation from the identity on Pairs to ι • R. Clearly if S is reduced, then S red = S, and there is trivially a natural transformation from R • ι to the identity on Pairs red .It is now elementary to check that these two natural transformations give the claimed adjunction.
As a consequence of reduction being a left adjoint functor, it commutes with colimits.
4.4.Quotients of pairs.Let S = (S 1 → S 0 ) be a pair.A quotient of S is a morphism of pairs such that both f 1 and f 0 are surjective.The kernel of f 1 is a differential congruence ker f 1 on S 1 , the kernel of f 0 is a congruence ker f 0 on S 0 , and π sends ker f 1 into ker f 0 .Conversely, a pair of congruences (K We now describe an important class of quotients.Suppose we are given a pair S = (S 1 → S 0 ) and a congruence K on the polynomial semiring S 0 {x 1 , . . ., x n } basic .By a slight abuse of notation, let (S 0 |S 1 ){x 1 , . . ., x n }/K denote the induced quotient, and then let S{x 1 , . . . ,x n } / / K denote the reduction of the pair S 1 {x 1 , . . ., x n } → (S 0 |S 1 ){x 1 , . . ., x n }/K.Quotients of this form will be used when we define the tropicalization of a system of differential equations in Section 5. → T 0 ) be a reduced S-algebra and K a congruence on S 0 {x 1 , . . ., x n } basic .Morphisms of S-algebras S{x 1 , . . . ,x n } / / K → T correspond bijectively with n-tuples y 1 , . . ., y n ∈ T 1 such that the elements π(d j y i ) ∈ T 0 define an S 0 -algebra homomorphism S 0 {x 1 , . . ., ) that are immediately seen to satisfy the stated condition.
Going in the other direction, suppose y i ∈ T 1 are elements satisfying the above condition.By Proposition 4.1.4,there is a uniquely determined morphism The images π(d j y i ) ∈ T 0 are equal to the elements f 0 (d j x i ) coming from S 0 {x 1 , . . ., x n } basic (recall that this is the polynomial algebra on the symbols d j x i ), and these define a semiring homomorphism S 0 {x 1 , . . ., x n } basic → T 0 .
that descends to the quotient by K, and since T is reduced, this descends to the reduction by Proposition 4.3.2.4.6.Colimits of pairs.In this section we show that colimits in the category of pairs can be computed by computing the colimits of the top and bottom individually.In order for this to be useful, it is helpful to note the following.
Proof.The category of idempotent semirings is cocomplete for the same reason as the category of rings; one can easily check that arbitrary coproducts and coequalizers exist.For differential idempotent semirings, one must only verify that tropical differentials d i on S i induce a tropical differential on the coproduct i S i , and likewise for coequalizers.Both of these verifications are elementary and straightforward.

Pairs
Semirings DiffSemirings commute with colimits, and π t also commutes with limits.
Proof.It suffices to show that π b admits a right adjoint and π t admits both a left and a right adjoint.
We start with π t .Let Finally, note that since the reduction functor is idempotent and has a left adjoint (Proposition 4.3.2), the colimit of a diagram of reduced pairs is reduced.4.7.Differential enhancements of seminorms.Given a seminorm v on R as in Definition 2.3.1, v(x) does not in general determine the seminorm of derivatives of x.In order to define tropical differential equations, we must enhance the seminorm with some additional information in order to determine the seminorms of sequences a, da, d 2 a, . ... To this end, we now introduce differential enhancements of seminorms.Definition 4.7.1.Given a differential ring R and a non-archimedean seminorm v : R → S 0 , a differential enhancement of v is a reduced pair S = (S 1 → S 0 ) and a map of sets v : R → S 1 such that (1) v(0) = 0 ∈ S 1 and v(1) = 1 ∈ S 1 ; (2) it commutes with the differentials: (3) the following diagram commutes: We will use the term differentially enhanced seminorm v = (v, v) : A → S to mean a seminorm v together with a differential enhancement v.
Note that if (v, v) : A → S is a differentially enhanced seminorm and ( f 0 , f 1 ) : S → T is a morphism of pairs, then the composition is also a differentially enhanced seminorm.

Example 4.7.2. Let k be a field and consider the differential ring of formal power series k[[t]] with differential d/dt. The t-adic norm k[[t]] → T admits a differential enhancement
This is the differentially enhanced seminorm used by Grigoriev [11] in his framework and subsequent works [1,4,6,7].While a seminorm u : R → S 0 may admit multiple distinct differential enhancements u, as illustrated in the examples above, it turns out that there is at most one for any given reduced pair S over S 0 .Proof.Consider the congruence K on S 1 generated by the relations u(x) ∼ u ′ (x) for x ∈ R. Since u and u ′ both commute with the differentials, K is a differential congruence, and since π • u = π • u ′ , it follows that K ⊂ ker π.Now, since S 1 → S 0 is reduced, K must be trivial.
In a differential ring R, an element a ∈ R is said to be a constant if d(a) = 0.The constants form a subring of R.
hold in S 0 , and hence the semiring homomorphism S 1 → S 0 factors through the quotient semiring S 1 /K because each of the generators of K (as a semiring congruence) is a relation that holds in S 0 .Since v commutes with the differentials, v(1), v(−1), v(a), v(b), v(a + b) and v(ab) are each constants in S 1 .From this we see that K is in fact a congruence of differential semirings.If K were nontrivial then the factorization S 1 → S 1 /K → S 0 would contradict the fact that S 1 → S 0 is reduced.Thus the equalities 4.8.The differential Berkovich space.Let k be a ring with a non-archimedean seminorm v : k → T. Given a k-algebra A, recall that the Berkovich analytification of Spec A is the set of seminorms w : A → T that are compatible with v in the sense that the composition k → A w → T is equal to v. The analytification is denoted (Spec A) an .It is equipped with a topology that we will not discuss here.
We now propose a generalization to the differential setting.Suppose k is a differential ring equipped with a differentially enhanced seminorm to S = (S 1 → S 0 ), and let A be a differential k-algebra.Given an S-algebra T, a differentially enhanced seminorm w = ( w, w) is said to be compatible with v if the diagram commutes.
Definition 4.8.1.Given an S-algebra T, the differential Berkovich space of A, denoted Berk T (A), is the set of differentially enhanced seminorms w : A → T that are compatible with v.
Note that there is a natural map Berk T (A) → (Spec A) an induced by sending a differentially enhanced seminorm w = ( w, w) to its underlying ordinary seminorm w.This map is injective thanks to Proposition 4.7.4.

TROPICALIZATION
5.1.Review of the non-differential case.In the familiar non-differential setting, one starts with a field k with a non-archimedean seminorm v : k → T, and one then defines three tropicalization maps: (1) Tropicalization of points is the map trop : k n → T n given by applying the seminorm coordinate-wise.

. , x n ]
given by applying the seminorm coefficient-wise.This extends to a map sending ideals in k[x 1 , . . ., x n ] to ideals in T[x 1 , . . ., x n ].
(3) Tropicalization of varieties sends V(I) to the subset of T n defined by the intersection of the tropical hypersurfaces of all f ∈ trop(I) x n ] is of course a system of polynomial equations, and a solution to this system is the same as a homomorphism k[x 1 , . . ., x n ]/I → k (or a homomorphism to some k-algebra A).Since the tropical hypersurface of a tropical polynomial f is exactly the solution set of the bend relations of trop( f ), it follows that the tropicalization of a variety is the set of solutions to the bend relations of the tropicalization of its defining ideal.Moreover, solutions to these bend relations are precisely homomorphisms of semirings T[x 1 , . . ., x n ]/Btrop(I) → T. One can thus think of the semiring T[x 1 , . . ., x n ]/Btrop(I) as the coordinate algebra of the tropical variety, and hence tropicalization of varieties has an incarnation at the level of algebras given by k[x 1 , . . . ,x n ]/I → T[x 1 , . . ., x n ]/Btrop(I) Note that this is a construction carried out when the seminorm v takes values in an idempotent semiring, not just T; see [9] for further details.5.2.Differential tropicalization.We now turn to the differential setting.Let k be a differential ring equipped with a differentially enhanced seminorm where x ǫ denotes a differential monomial in the variables x i .We have Proof.Given a morphism of presentations , and so it follows from Proposition 3.4.2that there is a functorially induced morphism of S-algebras defined by sending x a to a.It takes a formal differential polynomial in the elements of A and evaluates it to an element of A using the differential algebra structure of A. A similar morphism was studied in the non-differential setting in [8].In light of the following fact, we call this the universal presentation of A. Proof.Part (1): Let α : k{y i | i ∈ Λ} ։ A be a presentation.We will show that the set of morphisms Hom Pres(A) (α, Univ) contains exactly one element that is linearly monomial.Any morphism of presentations f from α to Univ must send each variable y i to a monomial in k{x a | a ∈ A} that is mapped to α(y i ) by Univ.One option is f (y i ) = x α(y i ) , and this is evidently the unique choice that defines a linearly monomial morphism.
Part (2): By (1), any finite presentation α admits a unique linearly monomial morphism α → Univ, and hence there is a canonical morphism u : colim ι → Univ.Given a finite presentation α : k{y 1 , . . ., y n } → A and an element a ∈ A, we extend to a new finite presentation α ′ : k{y 1 , . . ., y n , y a } → A by y a → a.The morphism α ′ → Univ sends y a to x a , and thus any element x a in the universal presentation is in the image of some finite presentation, so u is surjective.
We turn now to injectivity of u.If are two finite presentations with α(y 1 ) = β(z 1 ), then they each map to the presentation k{w, y 2 . . ., y n , z 2 , . . ., z m } by y 1 , z 1 → w and identity of all of the other generators.Hence u is injective as well.
Example 5.4.2.Consider the presentation α : k{y 1 , y 2 } → k{z} = A given by α(y 1 ) = z and α(y 2 ) = zdz.In addition to the linearly monomial morphism from α to Univ, there are also the monomial morphisms given by sending y 2 to x z x dz or x z dx z .
5.5.The universal tropicalization.We define the universal tropicalization of A to simply be the tropicalization of the universal presentation, trop(Univ).Given any element a ∈ A, any finite presentation α : k{x 1 , . . ., x n } → A maps to a finite presentation α ′ : k{x 1 , . . ., x n , x a } → A, with α ′ (x a ) = a.Hence it follows from Proposition 4.6.2 that j 1 and j 0 are both surjective.We turn to injectivity.
Observe that the congruence Btrop(ker Univ) on S 0 {x a | a ∈ A} is the transitive closure of the symmetric semiring generated by the bend relations of elements in trop(ker Univ), and so for any relation ( f ∼ g) ∈ Btrop(ker Univ) there exists a finite subset Λ ⊂ A containing all variables appearing in either f or g and such that, for the restriction Univ| Λ : k{x a | a ∈ Λ} → A, we have ( f ∼ g) ∈ Btrop(ker Univ| Λ ).
If Λ does not generate A as a differential algebra then we may add finitely many elements so that it does.We thus have a finite presentation β such that ( f ∼ g) is in the image of the canonical map Btrop(ker β) → Btrop(ker Univ).
Therefore the map j 0 : colim We now come to our main result, which says that a differential algebra A admits a universal differentially enhanced seminorm which is valued in the tropicalization of the universal presentation of A, (c.f.[8, Theorem A]).Hence, by Proposition 4.4.1, there is a unique morphism of S-algebras f w : trop(Univ) → T sending x a to y a on top, and to y a on the bottom.By construction, composition with f w sends u to w, and this completes the proof.

Definition 2 . 4 . 1 .
A tropical differential semiring is an idempotent semiring equipped with a tropical differential.A morphism of tropical differential semiringsf : (S, d S ) → (T, d T ) is a morphism of semirings such that f (d S (x)) = d T ( f (x)) for all x ∈ S.Remark 2.4.2.Just as the strict Leibniz relations for 2-fold products imply the strict Leibniz relations for n-fold products, such as d(xyz) = (dx)yz + x(dy)z + xy(dz), it is possible to derive n-fold tropical Leibniz relations from the 2-fold tropical Leibniz relations.I.e., in a tropical differential semiring the bend relations of any expression

1 ∼( 3 )( 4 )
If there are two leaves with labels a, b ∈ R having the same parent in a tree, then we may replace these two leaves with a single leaf labelled ab. a b ∼ ab If a leaf with label r ∈ R has a univalent non-root parent then we may replace the leaf and its parent by a single leaf with label dr.r ∼ dr

Proof.
By the construction of Forest(S, n), any n-tuple (a 1 , . . ., a n ) ∈ (S ′ ) n determines a homomorphism of semirings Forest(S, n) → S ′ that commutes with d, and since the tropical Leibniz relations hold in S ′ , this homomorphism descends to the quotient by B L trop .Conversely, a homomorphism provides an n-tuple of elements of S ′ .

Proposition 4 . 3 . 2 .
Sending S to S red defines a functor R : Pairs → Pairs red , and the quotient map S → S red is a natural transformation Id → R. Moreover R is left adjoint to the inclusion ι : Pairs ֒→ Pairs red .

4. 5 .Proposition 4 . 5 . 1 .
Solutions as morphisms.It follows directly from Proposition 4.4.1 and the definition of solutions to tropical differential equations that S-algebra morphisms S{x 1 , . . . ,x n } / / B(E) → S are in bijection with the solution set Sol(E).In fact, we have The functor S-alg → Sets sending an S-algebra S u → T to Sol(u * E) is corepresented by S{x 1 , . . ., x n } / / B(E).
L t : DiffSemirings → Pairs be the functor sending a differential idempotent semiring S to the pair S id → S, and let R t be the functor sending S to the pair S → * , where * denotes the trivial idempotent semiring consisting of a single element.Given a pair A p → B, a morphism of differential idempotent semirings f : S → A uniquely determines, and is uniquely determined by, a morphism of pairs that is evidently natural in the semiring S and the pair A → B. Thus L t is left adjoint to π t .For R t , observe that a morphism of differential semirings f : A → S is equivalent to a morphism of pairs: For π b , we will construct a right adjoint R b .Consider the subcategory Pairs/T ⊂ Pairs of pairs S → T, where a morphism is a morphism of pairs that is the identity on T. The colimit colim Pairs/T π t comes with a natural semiring homomorphism to T, and this defines a pair R b (T).It is straightforward to verify that R b (T) is functorial in T. A morphism of pairs A colim Pairs/T π t B T p f clearly provides a semiring homomorphism B → T. Conversely, given a semiring homomorphism B → T, the composition A → B → T is an object of Pairs/T and hence it has a canonical map to R b (T).

Proposition 4 . 7 . 4 .
Let u : R → S 0 be a seminorm and S 1 π → S 0 a reduced pair.If u, u ′ : R → S 1 are two differential enhancements of u, then u = u ′ .

Proposition 4 . 7 . 5 .
Given a differentially enhanced seminorm v = ( v : R → S 1 , v : R → S 0 ) on R, v restricts to a seminorm on the subring of constants in R.Proof.Suppose a, b are constants and consider the semiring congruence K on S 1 generated by the relations

Theorem 5 . 5 . 2 .
Given a differential k-algebra A, there is a differentially enhanced seminorm u = (u, u) : A → trop(Univ)sending d j x a → y (j)a descends to the quotient by the congruence Btrop(ker Univ).
1) For any morphism of idempotent semirings ν : S → T we have a pair (S, d = 0) → T, and it is reduced if and only if ν is injective.If ν is not injective, then we can replace S with im(ν) to obtain a reduced pair.defined by t n → e −n , and endow B[[t]] with the differential t n → t n−1 .This is a pair, and it is reduced by the following argument.Suppose a = b ∈ B[[t]].
1)We tropicalize points p ∈ k n via the map trop : k n → S n : k{x 1 , ..., x n } → S 0 {x 1 , ..., x n } basic .We write trop(I) for the ideal generated by the image of I, and so there is an induced a map sending ideals in k{x 1 , . .., x n } to ideals in S 0 {x 1 , . .., x n } basic .(3)We use the tropicalization of equations map to define a construction sending quotients α : k{x 1 , . . ., x n } ։ k{x 1 , . . ., x n }/I to quotients trop(α) of the pair S{x 1 , . . ., x n }.
trop Define trop(α) := S{x 1 , ..., x n } / / Btrop(I), where Btrop(I) is the congruence on (S 0 |S 1 ){x 1 , ..., x n } generated by the bend relations of trop(I) and we use the quotient construction at the end of Section 4.4.Remark 5.2.1.Since (S 0 |S 1 ){x 1 , . .., x n } is not a polynomial algebra, we cannot form the bend relations of an arbitrary element in it.The above construction uses the fact that applying v coefficient-wise lands in S 0 {x 1 , . .., x n } basic ⊂ (S 0 |S 1 ){x 1 , . .., x n }, and this algebra is a polynomial algebra so we can form bend relations in it.Proposition 5.2.2.Given a differential ideal I ⊂ k{x 1 , . .., x n }, tropicalization of points trop : k n → S n 1 sends Sol(I) into Sol(trop(I)).Proof.It suffices to show that if p ∈ k n is a solution to f ∈ k{x 1 , . . ., x n }, then trop vanishes.Since the differential enhancement map v commutes with the differentials, we have that v( f ǫ p ǫ ) is equal to v( f ǫ )π( v(p) ǫ ), which is equal to the evaluation of the differential monomial v( f ǫ )x ǫ at the point trop(p).Thus trop( f ) tropically vanishes at trop(p).Tropicalization of differential equations sends a presentation of a differential algebra to a tropical pair.Here we show that this defines a functor from a category of presentations to the category of tropical pairs.A homomorphism of differential algebras f : k{x 1 , . .., x n } → k{y 1 , . .., y m } is said to be monomial if each variable x i is sent to a monomial (with coefficient 1) in the variables y i and their derivatives (it need not send monomials to monomials in general), and it is said to be linearly monomial if each x i is sent to some y j , a derivative of y j , or to 0. The map f : k{x} → k{y} given by f (x) = ydy is monomial even though it sends the monomial dx to (dy) 2 + yd 2 y.However, it is not linearly monomial.We now define the category of presentations Pres(A).Objects of this category are presentations of A; i.e., an object is a Ritt algebra k{x i |i ∈ Λ} together with a surjective homomorphism k{x i |i ∈ Λ} → A, and whose morphisms are commutative triangles (5.3.1)k{x 1 , . . . ,x n } k{y 1 , . . . ,y m } ǫ v( f ǫ p ǫ ),tropically 5.3.Functoriality of tropicalization.f where f is a monomial morphism.We allow the set Λ of variables to be infinite, and let Pres fin (A) denote the subcategory of finite presentations.Let Pres lin (A) ⊂ Pres(A) denote the subcategory of presentations and linearly monomial morphisms.Proposition 5.3.2.The tropicalization construction (k{x i | i ∈ Λ} α → A) → trop(α) yields a functor Pres(A) → S-alg.