From Algebra to Analysis: New Proofs of Theorems by Ritt and Seidenberg

arXiv:2107.03012v1 [math.AC] 7 Jul 2021 h leri hoyo ifrnileutos loknown also equations, differential [ bra of theory algebraic The Introduction 1 eeta ed akt h el faayi.Teeaetoc two purpose: are this There for analysis. used typically of algebra realm differential the to back fields ferential theoretic from [ e.g., results (see, interesting a tives yielding This fruitful derivations). fi be commuting differential to several or of derivation elements a to with etc) ab typically meromorphic, one this, (analytic, doing For geometry. algebraic and bra ol rmmdlter osuydfeeta ed se e. (see, fields differential study to theory model from tools [email protected] [email protected] [email protected] 3 2 1 nti otx,afnaetlqeto shwt rnfrre transfer to how is question fundamental a context, this In aut fMcaisadMteais ocwSaeUnivers State Moscow Mathematics, and Mechanics of Univers Faculty State Moscow Mathematics, CNRS, LIX, and Mechanics of Faculty 19 ai at rmdfeeta ler n h lsia Cau classical the PDEs. and for theorem algebra differential from facts basic (Riquier’s analysis fr and tools theory) sophisticated set analysis use (characteristic of gebra results realm these the of to proofs PDEs existing nonlinear a on connect to results allowing theoretic algebra differential in results sical rmagbat nlss e rosof proofs new analysis: to algebra From ,am tsuyn olna ifrnileutosusing equations differential nonlinear studying at aims ], nti ae,w ienwsotpof o ohterm rely theorems both for proofs short new give we paper, this In theore embedding Seidenberg’s and zeroes of theorem Ritt’s .Pavlov D. hoesb itadSeidenberg and Ritt by theorems cl oyehiu,IsiuePltcnqed ai,Pa Paris, de Polytechnique Institute Polytechnique, Ecole ´ 2 , 5 , 6 , 1 16 , 21 ) utemr,oecnadtoal s powerful use additionally can one Furthermore, ]). .Pogudin G. Abstract 1 2 Yu.P. Razmyslov xsec theorem). existence ladapidperspec- applied and al gbacadmodel- and lgebraic mcntutv al- constructive om . [ g., chy-Kovalevskaya ls(ed equipped (fields elds tat rmfunctions from stracts sdfeeta alge- differential as poc undout turned pproach asclterm in theorems lassical oee,the However, . 14 asa,Fac,email: France, laiseau, t,Rsi,e-mail: Russia, ity, e-mail: Russia, ity, ehd falge- of methods , n nyon only ing r clas- are m ut bu dif- about sults 15 ]). 3 • Ritt’s theorem of zeroes [19, p. 176] which can be viewed as an analogue of Hilbert’s Nullstellensatz. The theorem implies that any system of nonlinear PDEs having a solution in some differential field has a solution in a field of meromorphic functions on some domain.

• Seidenberg’s embedding theorem [20] which is often used as a differential analogue of the Lefschetz principle (e.g. [1, 3, 6, 7, 10]). The theorem says that any countably generated differential ﬁeld with several commuting derivations can be embedded into a ﬁeld of meromorphic functions on some domain.

In [20], Seidenberg gave a complete proof of his theorem for the case of a single derivation (see also [14, Appendix A]). For the PDE case, he gave a sketch which reuses substantial parts of Ritt’s proof of Ritt’s zero theorem from [19]. The latter proof concludes the whole monograph and heavily relies on the techniques developed there. In particular, Ritt’s proof uses the machinery of characteristic sets [19, Chapter V] which is a fundamental tool in differential algebra but not so well-known in the broader algebra community and quite technical existence theorem for PDEs due to Riquier [19, Chapter VIII] (see also [18]) which, to the best of our knowledge, is not discussed in the standard PDE textbooks. Due to the importance of the theorems of Ritt and Siedenberg as bridges be- tween the algebraic and analytic theories of nonlinear PDEs, we think that it is highly desirable to have short proofs of these theorems accessible to people with some general knowledge in algebra and PDEs. In the present paper, we give such proofs. Our proofs rely only on some basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs. Our proof strategy is inspired by the argument from [8, Theorem 3.1] for the case of one derivation. However, the techniques from [8] had to be substantially developed in order to tackle the PDE case (which is quite subtle [13]) and to prove both Ritt’s and Seidenberg’s theorem (not only the Ritt’s as in [8]). The key ingre- dients of the argument are an auxiliary change of derivations (Lemma 1) which helps us to bring a system of PDEs into the form as in the Cauchy-Kovalevaskaya theorem, Taylor homomorpishms (Deﬁnition 7) allowing to build formal power series solutions, and a characterization of differentially simple algebras (Lemma 5). The paper is organized as follows. Section 2 contains the basic deﬁnitions used to state the main results in Section 3. Section 4 contains relevant notions and facts from algebra and analysis used in the proofs. The proofs are located in Section 5. Section 6 contains a remark on the special case of algebras over C.

2 Preliminaries

2.1 Algebra Throughout the paper, all algebras are assumed to be unital (that is, with a multi- plicative identity element).

2 m Notation 1 (Multi-indices). For every α =(α1,...,αm) ∈ Z>0 and for every tuple t =(t1,...,tm) of elements of a ring, we denote

α α1 αm t := t1 · ... · t and α! := α1! · ... · αm!.

Deﬁnition 1 (Differential rings and algebras). Let ∆ = {δ1,...,δm} be a set of symbols.

• Let R be a commutative ring. An additive map δ : R → R is called derivation if δ(ab)= δ(a)b + aδ(b) for any a, b ∈ R.

• A commutative ring R is called ∆-ring if δ1,...,δm act on R as pairwise commuting derivations. If R is a ﬁeld, it is called ∆-ﬁeld.

• Let A be a commutative algebra over ring R. If A and R are ∆-rings and the action of ∆ on R coincides with the restriction of the action of ∆ on R · 1A ⊆ A, then A is called ∆-algebra over R.

Deﬁnition 2 (Differential generators). Let A be a ∆-algebra over a ∆-ring R.A set S ⊆ A is called a set of ∆-generators of A over R if the set

α m {δ s | s ∈ S, α ∈ Z>0} of all the derivatives of all the elements of S generates A as R-algebra. A ∆- algebra is said to be ∆-finitely generated if it has a finite set of ∆-generators. ∆-generators for ∆-fields are defined analogously.

Deﬁnition 3 (Differential homomorphisms). Let A and B be ∆-algebras over ∆- ring R. A map f : A → B is called ∆-homomorphism if f is a homomorphism of commutative R-algebras and f(δa) = δf(a) for all δ ∈ ∆ and a ∈ A. An injective ∆-homomorphism is called ∆-embedding.

Definition 4 (Differential algebraicity). Let A be a ∆-algebra over a ∆-ring R. An α m element a ∈ A is said to be ∆-algebraic over R if the set {δ a | α ∈ Z>0} of all the derivatives of a is algebraically dependent over R. In other words, a satisfies a nonlinear PDE with coefficients in R.

2.2 Analysis Deﬁnition 5 (Multivariate holomorphic functions). Let U ⊆ Cm be a domain. A function f : U → C is called a holomorphic function in m variables on U if it is holomorphic on U with respect to each individual variable. The set of all holomorphic functions on U will be denoted by Om(U)

Notation 2. Let f be a holomorphic function on U ⊆ Cm. By V (f) we denote the set of zeroes of f.

Deﬁnition 6 (Multivariate meromorphic functions, [4, Chapter IV,Deﬁnition 2.1]). Let U ⊆ Cm bea domain. A meromorphic function on U is apair (f, M), where M is a thin set in U and f ∈ Om(U \ M) with the following property: for every z0 ∈ U,

3 there is a neighbourhood U0 of z0 and there are functions g, h ∈ Om(U0), such that V (h) ⊆ M and g(z) f(z)= for every z ∈ U \ M. h(z) 0

The set of meromorphic functions on a domain U is denoted Merm(U). By con- vention we define Mer0(U)= O0(U)= C. m For every domain U ⊆ C , the field Merm(U) has a natural structure of ∆- ∂ Cm field with δi ∈ ∆ acting as ∂zi , where z1,...,zm are the coordinates in . Fur- thermore, if U ⊆ V , then there is a natural ∆-embedding Merm(V ) ⊆Merm(U).

3 MainResults

Theorem 1 (Seidenberg’s embedding theorem). Let W ⊆ Cm be a domain and let K ⊆ Merm(W ) be at most countably ∆-generated ∆-field (over Q). Let L ⊃ K be a ∆-field finitely ∆-generated over K. Then there exists a domain U ⊆ W and a ∆-embedding f : L →Merm(U) over K. Theorem 2 (Ritt’s theorem of zeroes). Let W ⊆ Cm be a domain and let K ⊆ Merm(W ) be a ∆-field. Let A be a finitely generated ∆-algebra over K. Then there exists a non-trivial ∆-homomorphism f : A → Merm(U) for some domain U ⊆ W ⊆ Cm such that f(a) is ∆-algebraic over K for any a ∈ A. Corollary 1. Let A be a finitely ∆-generated ∆-algebra over C. Then there exists a m non-trivial ∆-homomorphism f : A → Om(U) for some domain U ⊆ C .

Proof. Ritt’s theorem yields the existence of a ∆-homomorphism f : A →Merm(W ). Let a1,...,an be a set of ∆-generators of A. There is a domain U ⊆ W such that f(a1),...,f(an) are holomorphic in U. Therefore, the restriction of f to U yields a ∆-homomorphism A → Om(U).

4 Notions and results used in the proofs

4.1 Algebra

Notation 3. Let R be a ∆-ring. By R[[z1,...,zm]] we denote the ring of formal power series over R in variables z1,...,zm. It has a natural structure of ∆-algebra ∂ over R with δi ∈ ∆ acting as ∂zi . Definition 7 (Taylor homomorphisms). Let A be a ∆-algebra over ∆-field K, L ⊇ K be a ∆-field and the action of ∆ on L be trivial. Let ψ : A → L be a (not necessarily differential) homomorphism of K-algebras. Let w ∈ Lm. Then we define a map called Taylor homomorphism Tψ,w : A → L[[t1,...,tm]] by the formula α α (t − w) Tψ,w(a) := ψ(δ a) for every a ∈ A. m α! αX∈Z>0

Direct computation shows [17, §44.3] that that Tψ,w is a ∆-homomorphism.

4 Notation 4. Let R be a ∆-ring. For every subset S ⊆ R, by ∆∞S we denote the α m set {δ s|α ∈ Z>0,s ∈ S} of all derivatives of the elements of S. Deﬁnition 8 (Differential polynomials). Let R be a ∆-ring. Consider an algebra of polynomials

∞ ∞ α m R[∆ x1,..., ∆ xn] := R[δ xi|α ∈ Z>0, i =1,...,n]

α in inﬁnitely many variables δ xi. We deﬁne the structure of ∆-algebra over R by

α α m δi(δ xj) := (δiδ )xj for every 1 6 i 6 m, 1 6 j 6 n, α ∈ Z>0.

The resulting algebra is called the the algebra of ∆-polynomials in x1,...,xn over R.

Deﬁnition 9 (Separants). Let R be a ∆-ring. Let P (x) ∈ R[∆∞x]. We introduce an ordering on the derivatives of x as follows:

α β δ x < δ x ⇐⇒ α

m µ where grlex is the graded lexicographic ordering of Z>0. Let δ x be the highest (w.r.t. the introduced ordering) derivative appearing in P . Consider P as a uni- µ α variate polynomial in δ x over R[δ x|α

Remark 1. Throughout the rest of the paper, we assume that the ordering of a set of derivatives of an element of a ∆-algebra is the one deﬁned in (1).

Deﬁnition 10 (Differential algebraicity and transcendence). Let R be a ∆-ring and let A be a ∆-algebra over R.

• A subset S ⊆ A is said to be ∆-dependent over R if ∆∞S is algebraically dependent over R. Otherwise, S is called ∆-independent over R.

• An element a ∈ A is said to be ∆-algebraic over R if the set {a} is ∆- dependent over R. Otherwise, a is called ∆-transcendental over R.

Definition 11 (Differential transcendence degree). Let A be a ∆-algebra over field K. Any maximal ∆-independent over K subset of A is called a ∆-transcendence basis of A over K. The cardinality of a ∆-transcendence basis does not depend on the choice of the basis [11, II.9, Theorem 4] and is called the ∆-transcendence degree ∆ of A over K (denoted by difftrdegK A). Definition 12 (Differential ideals). Let R be a ∆-ring. A subset I ⊆ R is called a differential ideal if it is an ideal of A considered as a commutative algebra and δa ∈ I for any δ ∈ ∆ and a ∈ I.

Notation 5. Throughout the rest of the paper, we use the notation ∆0 := ∆ \{δ1}.

5 4.2 Analysis The following is a special case of the Cauchy-Kovalevskaya theorem [9, Chapter V, §94] which is sufﬁcient for our purposes.

Theorem 3 (Cauchy-Kovalevskaya). Consider holomorphic functions in variables z1,...,zm. The operator of differentiation with respect to zi will be denoted by δi for i = 1,...,m. m For a positive integer r, we introduce a set of multi-indices Mr := {α ∈ Z>0 | |α| 6 r, α1

r α δ1u = F (z1,...,zm; δ u | α ∈ Mr), (2)

α where F is a rational function over C in z1,...,zm and derivatives {δ u | α ∈ Mr}. Consider complex numbers a1,...,am and functions ϕ0,...,ϕr−1 in variables z2,...,zm holomorphic in a neighbourhood of (a2,...,am) such that F is well-deﬁned under the substitution:

1. ai for zi for every 1 6 i 6 m

(α2,...,αm) α 2. and (δ ϕα1 )(a2,...,am) for δ u for every α ∈ Mr.

Then there is a unique function u holomorphic in a neighborhood of (a1,...,am) satisfy- ing (2) and i (δ1u)|z1=a1 = ϕi for every 0 6 i < r.

5 Proofs

This section is structured as follows. In Section 5.1, we introduce the notion of ∆-integral elements which is an algebraic way saying that an element satisﬁes a PDE as in the Cauchy-Kovalevskaya theorem. We prove that there always exists a linear change of derivations making a ﬁxed element ∆-integral (Lemma 1) and prove Lemma 2 which is a key tool for reducing the problem to the same problem in fewer derivations. Section 5.2 contains the proof of Seidenberg’s embedding theorem which pro- ceeds by induction on the number of derivations using Lemma 2. We deduce Ritt’s theorem of zeroes in Section 5.3 from Seidenberg’s theorem and Lemma 5 characterizing ∆-simple algebras.

5.1 Differentially integral generators Deﬁnition 13 (∆-integral elements). Let R be a ∆-ring and let A be a ∆-algebra over R. An element a ∈ A is said to be ∆-integral over R if there exists P (x) ∈ R[∆∞x] such that

∆ • P (a)=0, sepx (P )(a) =06 ;

r • the highest (w.r.t. the ordering (1)) derivative in P is of the form δ1x.

6 Remark 2. If a ∈ A is ∆-integral over R, then the equality δ1(P (a)) = 0 can be rewritten as

∆ r+1 α sepx P (a) · δ1 a = q(a), where q ∈ R[δ x | α

∆ r+1 q(a) Therefore, if sepx P (a) is invertible in A, we have δ1 a = ∆ . sepx (P )(a) Lemma 1. Let R be a ∆-ring and let A be a ∆-algebra over R. Let A be ∆-generated over R by ∆-algebraic over R elements a1,...,an. Then there exists an invertible Z- ∗ ∗ linear change of derivations transforming ∆ to ∆ such that a1,...,an are ∆ -integral over R.

Proof. Fix 1 6 i 6 n. Since ai is ∆-algebraic over R, there exists nonzero fi ∈ ∞ R[∆ x] such that fi(ai)=0. We will choose this fi so that its highest (w.r.t. (1)) derivative is minimal and, among such polynomials, the degree is minimal. We will call such fi a minimal polynomial for ai. We introduce variables λ2,...,λm algebraically independent over A and ex- tend the derivations from A to A[λ2,...,λm] by δiλj = 0 for all i = 1,...,m and j =2,...,m. Consider a set of derivations D := {d1, d2,..., dm} deﬁned by

d1 := δ1, dj := δj + λiδ1 for j =2, . . . , m.

Consider any 1 6 i 6 n. We rewrite fi in terms of D replacing δ1 with d1 and δj with dj −λi d1 for j = 2,...,m. We denote the order of the highest derivative ∂ appearing in fi by ri and the partial derivative ri fi by si. We will show that ∂(d1 x) si(ai) =06 . We write

∂fi q2 qm ∂fi si(x)= ri = λ2 ...λm q1 qm . ∂(d x) ∂(δ ...δm x) 1 q1+...X+qm=ri 1

Due to the minimality of fi as a vanishing polynomial of a and the algebraic independence of λj, the latter expression does not vanish at x = ai. So, si(ai) =06 . Since, for every 1 6 i 6 n, si(ai) is a nonzero polynomial in λ2,...,λm over ∗ ∗ A, it is possible to choose the values λ2,...,λm ∈ Z ⊂ R so that neither of si(ai) ∗ ∗ ∗ ∗ ∗ vanishes at (λ2,...,λm). Let ∆ = {δ1,...,δm} be the result of plugging these ∗ ∂f D sep∆ f(a )= i (a ) =06 i =1,...,n values to . Then we have x i ∗ ri i for every , ∂((δ1 ) x) ∗ so a1,...,an are ∆ -integral over R. Lemma 2. Let R be a ∆-ring and let A be a ∆-algebra over R. Assume that A is a domain and is ∆-generated by a1,...,an over R which are ∆-integral over R. Then there exists a ∈ A such that A[1/a] is finitely ∆0-generated over R. Proof. We will prove the lemma by induction on the number n of ∆-generators of A. If n =0, then A = R and A is clearly finitely ∆0-generated. Assume that the lemma is proved for all extensions ∆-generated by less than n ∞ ∞ elements. Applying the induction hypothesis to ∆-algebra A0 := R[∆ a1,..., ∆ an−1], we obtain b0 ∈ A0 such that A0[1/b0] is a finitely ∆0-generated R-algebra.

7 ∞ Since an is ∆-integral over R, there exists P (x) ∈ R[∆ x] such that P (an)=0, r ∆ the highest derivative in P is δ1x, and b2 := sepx (P )(an) =06 . We claim that

1 1 ∞ 1 ∞ 6r A = A0 , ∆0 , ∆0 (δ1 an) , (3) b1b2 b1 b2

6r r where δ1 an := {an, δ1an,...,δnan}. Since A0[1/b1] is finitely ∆0-generated over R, this would imply that A[1/(b1b2)] is finitely ∆0-generated over R as well. 6r In order to prove (3), it is sufficient to show that the images of {δ1 an, 1/b2} ∞ ∞ 6r

r+1 −q(an) −q(an) ∞ 6r δ1 an = ∆ = ∈ B, where q ∈ R[∆0 (δ1 x)]. sepx (P )(an) b2

• For δ1(1/b2), we observe that

1 1 ∞ 6r δ1 ∈ 2 A0[∆0 (δ1 an)] ⊆ B. b2 b2

5.2 Proof of Seidenberg’s Theorem Lemma 3. Let W ⊆ Cm be a domain and K be a countably ∆-generated subfield of Merm(W ). Then there exist c ∈ C and a domain V ⊆ W ∩{z1 = c} such that, for every f ∈ K, f|{z1=c} is a well-defined element of Merm−1(V ) and, therefore, the restriction to {z1 = c} defines a ∆0-embedding K →Merm−1(V ). ∞ Proof. Let K be ∆-generated by {bi}i=1. For every i > 0, denote by Si the set of m singularities of bi. By definition, Si is a nowhere dense subset of C . Therefore, ∞ m the union S = Si is a meagre set. As W is a domain in C , the difference W \ S iS=1 is non-empty. Choose any point (w1,...,wm) ∈ W \ S. Then all the restrictions of bi to {z1 = w1} are holomorphic at (w2,...,wm) and meromorphic in some vicinity V ⊆ W ∩{t1 = w1} of (w2,...,wm). Since every element f ∈ K is a rational function in bi’s and their partial derivatives, its restriction to {z1 = w1} is also a well-defined meromorphic function on V . Lemma 4. Let U ⊆ Cm be a domain. For every countably ∆-generated ∆-field K ⊆ ∆ Merm(U), difftrdegK Merm(U) is infinite. ∆ Proof. Suppose difftrdegK Merm(U) = l < ∞. Let τ1,...,τl be a ∆-transcendence basis of Merm(U) over K. Let L be a field ∆-generated by K and τ1,...,τl. Note that L is still at most countably ∆-generated and Merm(U) is ∆-algebraic over L. Choose an arbitrary point c ∈ U and denote by F a subfield of C generated by the values at c of those elements of L that are holomorphic at c. Clearly F is at most countably generated and the transcendence degree of C over F is infinite. Now any function in Merm(U) that is holomorphic at c and such that the set of values of its derivatives at c is transcendental over F is ∆-transcendental over L, which contradicts to the assumption that Merm(U) is ∆-algebraic over L.

8 Notation 6. Let A be a ∆-algebra without zero divisors. By Frac(A) we denote the ﬁeld of fractions of A.

We are now ready to prove Seidenberg’s theorem.

Proof. We will ﬁrst reduce the theorem to the case when L is ∆-algebraic over K. Assume that it is not, and let u1,...,uℓ be a ∆-transcendence basis of L over K. Lemma 4 implies that there exist functions f1,...,fℓ ∈ Merm(W ) ∆- transcendental over K. Let K˜ be a ∆-ﬁeld generated by K and f1,...,fℓ. The embedding K → L can be extended to an embedding K˜ → L by sending fi to ui for every 1 6 i 6 ℓ. Therefore, by replacing K with K˜ we will further assume that L is ∆-algebraic over K. We will prove the theorem by induction on the number m of derivations. If m =0, then L can be embedded into C by [12, Chapter V, Theorem 2.8]. Suppose m > 0. Let a1,...,an be a set of ∆-generators of L over K. Let ∞ ∞ A := K[∆ a1,..., ∆ an]. Since A is ∆-algebraic over K, by Lemma 1, there exist and invertible m × m matrix M over Q such that, for a new set of derivations

∗ ∗ ∗ ∆ = {δ1 ,...,δm} := M∆,

∗ ∗ a1,...,an are ∆ -integral over K. Due to the invertibility of M, every ∆ -embedding L →Merm(U) over K yields a ∆-embedding. Therefore, by changing the coordi- m −1 nates in the space C from (z1,...,zm) to M (z1,...,zm), we can further assume ∗ that ∆ = ∆ , so a1,...,an are ∆-integral over K. −1 Lemma 2 implies that there exists a ∈ A such that B := A[a ] is ﬁnitely ∆0- generated. By ∆-integrality of a1,...,an and Remark 2, for every 1 6 i 6 n, α there exists a positive integer ri and a rational function gi ∈ K(δ y | α

ri δ ai = gi(ai). (4)

Since K is at most countably ∆-generated, Lemma 3 implies that there ex- m−1 ist w1 ∈ C and V ⊆ W ∩{z1 = w1} ⊆ C such that the restriction to {z1 = w1} induces a ∆0-embedding ρ: K → Merm−1(V ). We apply the induction hypothesis to ∆0-ﬁelds ρ(K) and Frac(B) = L. This yields ∆0-embedding h : L →

Merm−1(V ) for some V ⊆ V . Choose a point v = (w ,...,w ) ∈ V such that all the h(a ) are holomorphic e e 2 m i at v and all the g (a ) are holomorphic at w = (w ,w ,...,w ) ∈ W . Consider i i e 1 2 m the Taylor homomorphism Th,w : A → Merm−1(V )[[z1]] deﬁned as follows (see Deﬁnition 7): e ∞ (z − w )k T (a) := h(δka) 1 1 for every a ∈ A. h,w 1 k! Xk=0

9 Note that Th,w is a ∆-homomorphism. Fix 1 6 i 6 n. Since a1 is a solution of (4) and Th,w is a ∆-homomorphism, Th,w ri is a formal power series solution of δ1 y = gi(y) corresponding to holomorphic initial conditions

ri−1 ri−1 y|z1=w1 = h(ai), (δ1y)|z1=w1 = h(δ1ai),..., (δ1 y)|z1=w1 = h(δ1 ai).

By the Cauchy-Kovalevskaya theorem, this solution is holomorphic in some vicin- n ity Ui of w. We set U := i=1 Ui. Thus, Th,w induces a non-trivial ∆-homomorphism from A to Merm(U). SinceT h is injective, Th,w is also injective, so it can be extended to a ∆-embedding L →Merm(U) over K.

5.3 Proof of Ritt’s theorem Deﬁnition 14 (Differentially simple rings). A ∆-ring R is called ∆-simple if it contains no proper ∆-ideals.

Lemma 5. Let A be a ∆-simple ∆-algebra ∆-generated by a1,...,an over a ∆-ﬁeld K. Then A does not contain zero divisors. Furthermore, assume that there exists an integer ℓ such that

• a1,...,aℓ form a ∆-transcendence basis of A over K;

∞ ∞ • aℓ+1,...,an are ∆-integral over K[∆ a1,..., ∆ aℓ].

Then A has ﬁnite ∆0-transcendence degree over K. In particular, ℓ =0.

Proof. Consider any non-zero (not necessarily differential) homomorphism ψ : A → F (F ⊇ K is a field) and the corresponding Taylor homomorphism Tψ,0 : A → F [[z1,...,zm]], which is a ∆-homomorphism. Since A is ∆-simple, the kernel of Tψ,0 is zero. Therefore, Tψ,0 is a ∆-embedding of A into F [[z1,...,zm]]. Since the latter does not contain zero divisors, the same is true for A. Assume that A has infinite ∆0-transcendence degree, that is, ℓ > 0. Since ∞ ∞ aℓ+1,...,an are ∆-integral over R := K[∆ a1,..., ∆ aℓ], Lemma 2 implies that there exists an element b ∈ A such that A0 := A[1/b] is a finitely ∆0-generated ∞ ∞ algebra over R. Note that A0 is also ∆-simple. Let A0 = R[∆0 b1,..., ∆0 bs]. For every j > 0, consider ∆0-algebra

∞ (

For every j > 0, we have

∆0 jl 6 diﬀtrdegK Bj 6 jl + s.

j j This inequality implies that there exists N such that, for every j > N, δ1a1,...,δ1al are ∆0-independent over Bj. Consider any non-zero ∆0-homomorphism

ϕ˜: BN → L,

10 where L ⊇ K is a ∆0-field. Due to the ∆0-independence of the rest of the ele- j ments δ1ai for 1 6 i 6 l and j > N, ϕ˜ can be extended to a homomorphism ϕ: A0 → j L so that ϕ(δ1ai)=0 for every 1 6 i 6 l and j > N. Consider a Taylor homomorphism Tϕ,0 : A0 → L[[z]] with respect to δ1. Since ϕ was a ∆0-homomorphism, Tϕ,0 is a ∆-homomorphism. It remains to observe N+1 N+1 that the kernel of Tϕ,0 contains δ1 a1,...,δ1 al contradicting to the fact that A0 is ∆-simple. Theorem 2 (Ritt’s theorem of zeroes). Let W ⊆ Cm be a domain and let K ⊆ Merm(W ) be a ∆-field. Let A be a finitely generated ∆-algebra over K. Then there exists a non-trivial ∆-homomorphism f : A → Merm(U) for some domain U ⊆ W ⊆ Cm such that f(a) is ∆-algebraic over K for any a ∈ A.

∞ ∞ Proof. We can represent A as A = R/J, where R := K[∆ x1,..., ∆ xn] and J ⊆ R is a differential ideal. Since R is a countable-dimensional K-space, J can be generated by at most countable set of generators. Pick any such set and denote the ∆-field generated by the coefficients of the generators by K0. Then K0 ∞ ∞ is countably ∆-generated. Let R0 := K0[∆ x1,..., ∆ xn]. Since J is defined over

K0, for A0 := R0/(J ∩ R0), we have A = K ⊗K0 A0. Let I be a maximal differential ideal of A0, and consider the canonical projec- tion π : A0 → A0/I. Let a1,...,an be a set of ∆-generators of A0/I. Since A0/I is differentially simple, by Lemma 5, A0/I does not have zero divisors and is ∆- algebraic over K0. We apply Theorem 1 to the ∆-ﬁelds K0 ⊆ Frac(A0/I) and obtain a ∆-embedding h: A0/I → Merm(U) over K. Since A0/I is ∆-algebraic over K0, h(a) is also ∆-algebraic over K0 for any a ∈ A0/I. Let f0 := h ◦ π, then f0(a) is ∆-algebraic over K0 for any a ∈ A0. Since K ⊆ Merm(U), we can construct a nontrivial ∆-homomorphism f : K ⊗K0 A0 →Merm(U) as the tensor product of the embedding K → Merm(U) and f. The ∆-algebraicity of the image of f0 over K0 implies the ∆-algebraicity of the image of the image of f over K.

6 Remarks on the analytic spectrum

Definition 15 (Analytic spectrum). Consider C as a ∆-field with the zero derivations. Let A be a finitely ∆-generated ∆-algebra over C. A homomorphism (not necessarily differential) ψ : A → C of C-algebras is called analytic if, for every a ∈ A, the formal power series Tψ,0(a) has a positive radius of convergence. The set of the kernels of analytic C-homomorphisms is called the analytic spectrum of A. Corollary 1 implies the following. Corollary 2. Let A be a finitely generated ∆-algebra with identity over C. Then the analytic spectrum of A is a Zarisky-dense subset of its maximal spectrum. Proof. Assume that analytic spectrum of A is not Zarisky-dense in spec A. Then it is contained in some maximal proper closed subset F = {M ∈ spec A|a ∈ M} for some non-nilpotent a ∈ A.

11 Consider the localization A[a−1] and a non-trivial ∆-homomorphism f : A[a−1] → Om(U) given by Corollary 1. Then f(a) =6 0. We ﬁx u ∈ U such that f(a)(u) =6 0 and consider a homomorphism g : A → C deﬁned by g(b) := f(b)(u) for every b ∈ A. Note that I := Ker(g) is an analytic ideal such that a 6∈ I, so we have arrived at the contradiction.

Acknowledgements GP was partially supported by NSF grants DMS-1853482, DMS-1760448, and DMS-1853650 and by the Paris Ile-de-France region.

References

[1] G. Binyamini. Bezout-type theorems for differential ﬁelds. Compositio Mathematica, 153(4):867–888, 2017. URL https://doi.org/10.1112/s0010437x17007035.

[2] F. Boulier and F. Lemaire. Differential algebra and system modeling in cel- lular biology. In Algebraic Biology, pages 22–39. Springer Berlin Heidelberg. URL https://doi.org/10.1007/978-3-540-85101-1_3.

[3] A. Buium. Geometry of differential polynomial functions, III: Mod- uli spaces. American Journal of Mathematics, 117(1):1, 1995. URL https://doi.org/10.2307/2375035.

[4] W.Fischer andI.Lieb. A Course in Complex Analysis. Vieweg+Teubner Verlag, 2012. URL https://doi.org/10.1007/978-3-8348-8661-3.

[5] M. Fliess. Nonlinear control theory and differential algebra: Some illus- trative examples. IFAC Proceedings Volumes, 20(5):103–107, 1987. URL https://doi.org/10.1016/s1474-6670(17)55072-7.

[6] J. Freitag and T. Scanlon. Strong minimality and the j-function. Jour- nal of the European Mathematical Society, 23(1):119–136, 2017. URL https://doi.org/10.4171/jems/761.

[7] H. Gauchman and L. A. Rubel. Sums of products of functions of x times functions of y. Linear Algebra and its Applications, 125:19–63, 1989. URL https://doi.org/10.1016/0024-3795(89)90031-1.

[8] O. V. Gerasimova, Y. P. Razmyslov, and G. A. Pogudin. Rolling simplexes and their commensurability. III (Capelli identities and their application to differential algebras). Journal of Mathematical Sciences, 221:315–325, 2017. URL https://doi.org/10.1007/s10958-017-3229-3.

[9] E. Goursat. A Course in Mathematical Analysis, volume II, part two. Dover Publications, Inc., 1945.

12 [10] C. Hardouin and M. F. Singer. Differential Galois theory of linear difference equations. Mathematische Annalen, 342(2):333–377, 2008. URL https://doi.org/10.1007/s00208-008-0238-z.

[11] E. Kolchin. Differential Algebra and Algebraic Groups. Academic Press, New York, 1973.

[12] S. Lang. Algebra. Springer-Verlag New York, Inc., 2002.

[13] F. Lemaire. An orderly linear PDE system with analytic initial conditions with a non-analytic solution. Journal of Symbolic Computation, 35(5):487–498, 2003. URL https://doi.org/10.1016/s0747-7171(03)00017-8.

[14] D. Marker, D. Marker, M. Messmer, and A. Pillay. Model theory of differential ﬁelds. In Model Theory of Fields, pages 38–113. Cambridge University Press. URL https://doi.org/10.1017/9781316716991.003.

[15] J. Nagloo. Model theory and differential equations. No- tices of the American Mathematical Society, 68(02):1, 2021. URL https://doi.org/10.1090/noti2221.

[16] J. Pila and J. Tsimerman. Ax-Schanuel for the j-function. Duke Mathematical Journal, 165(13), 2016. URL https://doi.org/10.1215/00127094-3620005.

[17] Y. P. Razmyslov. Identities of Algebras and their Representations. Translations of Mathematical Monographs. American Mathematical Society, 1994.

[18] C. Riquier. Les systèmes d’équations aux dérivées partielles. Gauthier-Villars, 1910.

[19] J. F. Ritt. Differential Algerba. 1950.

[20] A. Seidenberg. Abstract differential algebra and the analytic case. Pro- ceedings of the American Mathematical Society, 9(1):159–164, 1958. URL https://doi.org/10.2307/2033416.

[21] M. van der Put and M. F. Singer. Galois Theory of Linear Dif- ferential Equations. Springer Berlin Heidelberg, 2003. URL https://doi.org/10.1007/978-3-642-55750-7.