Completeness in partial differential algebraic geometry

Journal of Algebra 420 (2014) 350–372

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Journal of Algebra www.elsevier.com/locate/jalgebra

Completeness in partial differential algebraic geometry James Freitag Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720-3840, United States

a r t i c l e

i n f o

Article history: Received 23 August 2011 Available online 16 September 2014 Communicated by Leonard L. Scott, Jr. Keywords: Differential algebraic geometry Completeness Partial differential fields Model theory Model theory of fields Differentially closed fields

a b s t r a c t This paper is part of the model theory of fields of characteristic 0, equipped with m commuting derivation operators (DCF 0,m ). It continues to partial differential fields work begun by Wai-Yan Pong, who treated the case m = 1. We study the concept of completeness in differential algebraic geometry, applying methods of model theory and differential algebra. Our central tool in applying the valuative criterion developed in differential algebra by E.R. Kolchin, Peter Bloom, and Sally Morrison is a fundamental theorem in classical elimination theory due to the model theorist Lou van den Dries. We use this valuative criterion to give a new family of complete differential algebraic varieties. In addition to completeness, we prove some embedding theorems for differential algebraic varieties of arbitrary differential transcendence degree. As a special case, we show that every differential algebraic subvariety of the projective line which has Lascar rank less than ω m can be embedded in the affine line. © 2014 Elsevier Inc. All rights reserved.

The theory of differential fields of characteristic zero with finitely many commuting derivations δ1 , . . . , δm has a model companion, the theory of differentially closed fields, denoted by DCF 0,m . The theory is characterized by the property: F is differentially E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jalgebra.2014.07.025 0021-8693/© 2014 Elsevier Inc. All rights reserved.

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closed if and only if any system of partial differential equations over f which has a solution in some differential field extension of F already has a solution in F . Seeing that this property is first order is, of course, nontrivial. In this paper, we fix a large saturated enough model U |= DCF 0,m . In the notation of Kolchin, U is universal over F or any of the differential fields which we consider in this paper; for instance, one could assume without loss of generality that every differential subfield we consider in this paper is of cardinality less than κ and that U is κ-saturated [15, for a reference on the basics of saturation]. The model theory of partial differential fields was developed in [17], where the basic properties of this theory were originally investigated. Throughout this paper, all differential fields will be assumed to be subfields of U; this is only a matter of convenience and does not affect any of the results. Throughout the paper, we will let F denote a fixed arbitrary differentially closed subfield F of U. We mention briefly that because this theory DCF 0,m has quantifier elimination, the F -definable sets in Un correspond to the F -constructible sets in the Kolchin topology, a differential version of the Zariski topology where the closed sets are given by the vanishing of differential polynomials over F . This fixed Δ-field F will serve throughout the paper, and unless specifically noted, Δ-algebraic varieties are assumed to be defined over F , Δ-rings are Δ–F -algebras, and ring homomorphisms are morphisms of Δ–F -algebras. The differential varieties we consider will be given by closed subsets in affine or projective space; since we have fixed a large saturated model U throughout the paper, we are (wlog) considering points with coordinates in U. So, Pn means Pn (U) (similarly for affine space). When we are considering points with coordinates in some other field or ring R, we will use the notation Pn (R). Generalizations to differential schemes are of interest, but are not treated here, see [13,14,9, for instance]. Pillay also considers differential completeness for a slightly different category in [24]; the precise relationship between the notions is not clear, but completeness in the category of D-varieties seems more akin to the notion in the algebraic category. Generalizing this work to the differencedifferential topology is of interest, but there are important model-theoretic obstacles. Specifically, differential-difference fields do not have quantifier elimination [18]. Nevertheless, the difference-differential category is of particular interest because we know that the natural analogues of Questions 4.11 and 4.12 are actually known to be distinct. The definition of Δ-completeness 2.4 is a straightforward generalization of the definition of completeness in the category of abstract algebraic varieties (integral separated schemes of finite type over an algebraically closed field). Section 2 is devoted to giving the basic definitions and particular notation of this paper. After the basic examples, we turn to an example of Kolchin in Section 3, which highlights one of the essential differences between Δ-completeness and completeness for abstract algebraic varieties: Pn is not Δ-complete. We should note that this paper only considers quasiprojective differential algebraic varieties over a differentially closed field. The completeness question for more abstract differential varieties [12] is of interest. In the category of abstract algebraic varieties, there are complete varieties which are not projective

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[22,28, Chapter 6, Section 2.3]. We do not know about the existence of such complete differential schemes (which are non-projective in the Δ-topology). We also do not know of any projectivity criterion for complete differential schemes. For instance, in the setting of abstract algebraic varieties, see the Chevalley–Kleiman Criterion [28, Chapter 6, Section 2.4]. Even in the quasiprojective case, the notion of Δ-completeness sits in stark contrast to its algebraic counterpart. Pong [26] proved that all Δ-complete ordinary differential algebraic varieties are of finite transcendence degree (δ-transcendence degree zero). Using the model-theoretic tool of Lascar rank, one can prove that any finite transcendence degree ordinary differential algebraic variety is affine and may in fact be embedded in A1 . This fact blurs the lines between projective and affine differential algebraic varieties; projective differential algebraic varieties like the one given by   Z zy 2 − x3 − azx2 − bz 3 , zδ(x) − xδ(z), zδ(y) − yδ(z) ⊂ P2 are isomorphic to affine differential algebraic varieties, even subvarieties of A1 . In this paper, we generalize Pong’s results in several ways. The main generalization is the setting: we work with partial differential algebraic varieties. In this case, we prove the analogues of the results mentioned above; the Δ-complete varieties are of Δ-transcendence degree zero and may be embedded in A1 . We also prove several embedding theorems for positive Δ-transcendence degree differential algebraic varieties, which generalize Pong’s results in the ordinary case (he only considers varieties of δ-transcendence degree zero). A very simple instance of the embedding theorems discussed above will be slowly developed throughout the text via a series of Examples 2.3, 2.6, 4.13, where we consider the constant points of the above elliptic curve. The examples reveal some of the exotic nature of Δ-complete differential algebraic varieties as the interesting potential examples coming from the embedding theorems of Section 4 indicate. In Section 5, we turn to a criterion for proving differential algebraic varieties are complete. The first ingredient is a result of van den Dries [32], which relates sets definable by positive quantifier free formulas to homomorphisms of substructures (in differential fields, substructures are differential subfields): Proposition. T a complete L-theory and φ(v1 . . . vn ) an L-formula without parameters. Then the following are equivalent: 1. There is a positive quantifier free formula ψ such that T proves ∀vφ(v) ↔ ψ(v). 2. For any K, L |= T and f : A → L an embedding of a substructure A of K into L, if a ∈ Am and K |= φ(a) then L |= φ(f (a)). In the differential algebra, the definable sets in condition 1) above are Kolchin closed subsets of affine space. It is worth mentioning for the non-model theorist that assumption

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in the above proposition that φ is definable without parameters (the corresponding variety would then be over Q in the natural language of differential rings) is a red herring. One may add formal constants (not to be confused with constants of Δ – here constant is a model-theoretic term) to the language, whose interpretations in any model are precisely the parameters used in φ. The proposition applies equally well to the new theory T ∗ , the theory of any model of T in the language L extended by the new constant symbols. In the differential algebra, the Δ-ring-homomorphisms in condition 2) above were studied by Blum [1] and Morrison [21]. Thanks to the referee for pointing out [2], where the study of completeness in the differential setting was initiated. Morrison and Blum considered the problem of extending Δ-ring homomorphisms; in particular, they exhibit examples of Δ-ring homomorphisms which cannot be extended. This is in contrast to the case of (non-differential) fields where any valuation on a field F has an extension to a valuation on any extension field K/F . In the differential context, it is the domains of non-extendible Δ-ring homomorphisms which form the basis for our valuative criterion. Our valuative criterion for a variety V involves considering the R-rational points of the differential algebraic variety, where R is a maximal Δ-subring of a differentially closed field (a Δ-subring is maximal if it is the domain of a non-extendable Δ-ring homomorphism Definition 5.4). We do not establish any result regarding the nature of maximal Δ-subrings, but rather use them as a tool via the results of Blum and Morrison (providing examples showing their utility). Further results regarding maximal Δ-rings would be of interest, in part because they might be brought to bear on the completeness problem. On the other hand, further results on complete Δ-varieties would be of purely algebraic interest even if obtained via completely different methods, because they may shed light on the problem of extending differential homomorphisms. We should note that the approach we describe here has been taken up in much greater detail in various special cases over ordinary differential fields in some recent work of William Simmons [29]. In Section 6, we use the criterion to construct some new examples of Δ-complete varieties. When moving from the ordinary setting to the partial setting, there are two fairly natural generalizations of the finite transcendence degree differential algebraic varieties over ordinary differential fields: the finite transcendence degree partial differential algebraic varieties and the partial differential algebraic varieties with Δ-type less than m, the number of derivations. The Δ-type of a differential algebraic variety is the degree of its Kolchin polynomial [4]. For proving the results of this paper, the second generalization turns out to be the appropriate one. Some of the results have proofs which are straightforward generalizations of the analogous results from the ordinary case [26] once one sets up the correct definitions. We reproduce even those proofs here for several reasons. First, this keeps the paper more self-contained and shows the transition to the partial case once the assumptions and definitions are in place. Also, some of the results involve generalizations in multiple directions (for instance Proposition 4.4 involves moving from the ordinary to the partial case, as well as dropping the assumption of zero dimensionality). We also take the opportunity to explain some of the model-theoretic

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notation in more algebraic terms. Finally, we give some specific examples and provide additional discussion about some of the open questions regarding completeness in this category. 1. Notation In this paper, we will take F to be a differentially closed field of characteristic zero in m commuting derivations, Δ = {δ1 , . . . , δm }. Or, in model-theoretic terms, F |= DCF 0,m . All of the varieties which we consider will be defined over F , but of course they are always defined over some finitely generated Δ-field extension of Q in U. We use F mainly out of convenience because our results usually do not require being more sensitive and precise about the field of definition. In this paper, we do not consider abstract differential algebraic varieties or differential schemes; we only consider projective and quasiprojective differential algebraic varieties. As mentioned, we will often regard Δ–F -varieties as synonymous with their U-rational points (for more general notions of differential algebraic varieties, it may be useful to use a setup emphasizing that each differential algebraic variety is a representable functor in this category, but this is not necessary for our results). For instance, we will use the notation:   C = x ∈ U | δ1 (x) = . . . = δm (x) = 0 . We also sometimes consider the rational points of a variety V for other differential fields or rings; in this case, we write V (K) or V (R). Occasionally in constructions, we use morphisms and varieties defined over U. In these cases, as we will note, the model completeness of DCF 0,m allows us to construct morphisms and varieties with the same relevant (first order) properties defined over F . As mentioned above, these morphisms and closed sets can further be assumed to be defined over some finitely generated Δ-field extension of Q. The type of a finite tuple of U over F is the collection of all first order formulae with parameters from F which hold of the tuple. A realization of a type p over F (we write p ∈ S(F )) is simply a tuple from a field extension satisfying all of the first order formulae in the type. As DCF 0,m has quantifier elimination, we have a correspondence between types and prime differential ideals and irreducible differential algebraic varieties. Given a type p ∈ S(F ), we have a corresponding (prime, by the assumption that F is algebraically closed) differential ideal via   p → Ip = f ∈ F {y} | f (y) = 0 ∈ p . Of course, the corresponding variety is simply the zero set of Ip . For the reader not acquainted with the language of model theory, the type of a tuple b over F corresponds to the isomorphism type of the differential field extension F b /F (with specified

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generators). Next, we will give a series of definitions leading to one of the main tools used in this paper, Lascar rank, an ordinal valued function on types. αm Let Θ be the free commutative monoid generated by Δ. For θ ∈ Θ, if θ = δ1α1 . . . δm , then ord(θ) = α1 + . . . + . . . + αm . The order gives a grading on the monoid Θ. We let   Θ(s) = θ ∈ Θ : ord(θ) ≤ s . Let k be an arbitrary differential field. Theorem 1.1. (See [11, Theorem 6, page 115].) Let η = (η1 , . . . , ηn ) be a finite family of elements in some extension of k. There is a numerical polynomial ωη/k (s) with the following properties. 1. For sufficiently large s ∈ N, the transcendence degree of k((θηj )θ∈Θ(s),1≤j≤n ) over k is equal to ωη/k (s). 2. deg(ωη/k (s)) ≤ m. 3. One can write ωη/k (s) =

 o≤i≤m

 ai

 s+i . i

In this case, am is the differential transcendence degree of kη over k. Definition 1.2. The polynomial from the theorem is called the Kolchin polynomial or the differential dimension polynomial. Let p ∈ S(k). Then ωp (t) := ωb/k (t) where b is any realization of the type p. Remark 1.3. When the Kolchin polynomial is written in the canonical form given in part 3) of Theorem 1.1, the coefficients are integers which are not necessarily nonnegative. The degree of the Kolchin polynomial is called the Δ-type of the field extension and the leading coefficient is called the typical Δ-dimension. The Kolchin polynomial is not a differential birational invariant. However, the degree and leading coefficient are important integers that are not dependent on the choice of generators of a finitely generated differential field extension. The leading coefficient is always positive. Definition 1.4. Suppose that p and q are types such that q extends p. This means p ∈ S(k) for some differential field k and q ∈ S(K) where K is a differential field extension of k; further, as a sets of first order formulae p ⊂ q. In this case, any realization of q is necessarily a realization of p. Definition 1.5. Let q extend p. We say q is a nonforking extension of p if wp (t) = ωq (t).

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Definition 1.6. Let p be a type. Then, • RU (p) ≥ 0 if p is consistent. • RU (p) ≥ β, where β is a limit just in case RU (p) ≥ α for all α < β. • RU (p) ≥ α + 1 just in case there is a forking extension q of p such that RU (q) ≥ α. Remark 1.7. The last three definitions are specific instances of model theoretic notation in the setting of differential algebra; for the more general definitions, see [23]. More information about Lascar rank can be found in a variety of sources; in the case of differential fields, see [16,20]. When considering model-theoretic ranks on types like Lascar rank (denoted RU (p)) we will write RU (V ) for a definable set (whose Δ-closure is irreducible) for RU (p) where p is a type of maximal Lascar rank in V . We should note that some care is required, since the model theoretic ranks are not always invariant under taking Δ-closure. See [5] for an example. Sometimes it will be convenient to consider tuples in field extensions, in which case our notation using also uses the correspondence between definable sets, types, and tuples in field extensions. 2. Definitions and basic results A subset of An is Δ-closed if it is the zero set of a collection of Δ-polynomials in n variables. We use F {y1 , . . . , yn } to denote the ring of Δ-polynomials over F in y, . . . , yn . When we want to emphasize that the collection of Δ-polynomials is over F , we say Δ–F -closed. For a thorough development, see [11] or [16]. Definition 2.1. A Δ-polynomial in F {y0 , . . . , yn } is Δ-homogeneous of degree d if f (ty0 , . . . , tyn ) = td f (y0 , . . . , yn ), where t is a Δ–F -indeterminate. Remark 2.2. The reader should note that Δ-homogeneity is a stronger notion than homogeneity of a differential polynomial as a polynomial in Θx. For instance, δx − x is a homogeneous Δ-polynomial, but not a Δ-homogeneous Δ-polynomial. The reader may verify that the following is a Δ-homogeneous Δ-polynomial: yδx − xδy − xy.

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The vanishing of Δ-homogeneous Δ-polynomials in n + 1 variables is well-defined on Pn . Generally, we can easily homogenize an arbitrary Δ-polynomial in x with respect to a new variable y. Homogenization in the partial differential case works identically to the ordinary case. For more details and examples see [26]. However, next we will develop one specific example which will be used throughout this paper. Example 2.3. For this example, we assume that the differential field is ordinary with Δ = {δ}. We wish to consider, as a differential variety, the constant points of an elliptic curve. That is, we are considering the projective closure of:   V = Z y 2 − x3 − ax + b ⊆ A2 (C), where 4a3 +27b2 = 0 and the coefficients a, b are themselves constants. Here and throughout the paper, Z(P ) denotes the zero set of the differential polynomial P ; this notation is also used when a set of differential polynomials appears rather than a single differential polynomial. As a differential variety, V is the zero set of the above equation and δ(x) = δ(y) = 0. The projective closure of V in P2 is given by the zero set of the differential homogenizations of the three above equations. Of course, the algebraic equation is homogenized in the standard way: zy 2 = x3 + az 2 x + bz 3 . When f (x, y) ∈ F {x, y} is a differential polynomial in x and y, then for some sufficiently large d ∈ N, 

x y , z f z z



d

is differentially homogeneous of degree d. When f is irreducible over F , the minimal such d produces a homogeneous irreducible differential polynomial. In the case f (x, y) = δ(x), z2δ

  x = zδ(x) − xδ(z) z

is differentially homogeneous of degree 2. Similarly, the differential homogenization of δ(y) produces the degree 2 differential polynomial zδ(y) − yδ(x). The projective closure of V in P2 is the locus of the three equations. We will let   E := Z zy 2 − x3 − az 2 x − bz 3 , zδ(x) − xδ(z), zδ(y) − yδ(z) ⊂ P2 . This differential algebraic variety will be used as an example several times in the coming sections. We should note that calculating the projective closure of an affine differential algebraic variety can be slightly tricky. For instance, in the above variety, the hyperplane at infinity contains only the point [0 : 1 : 0], so it is not necessary to include the

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differential equation xδ(y) − yδ(x) = 0. On the other hand, consider the projective closure of   A2 (C) = (x, y) | δ(x) = δ(y) = 0 . In this case, the projective closure is given by   P2 (C) = Z zδ(x) − xδ(z), zδ(y) − yδ(z), xδ(y) − yδ(x) . Failing to include the last differential polynomial would result in including the entire line at infinity in P2 instead of only the constant points, that is, P2 (U) rather than P2 (C). In general, the Δ-closed subsets of Pn are the zero sets of collections of homogeneous differential polynomials in F {y0 , . . . , yn }. Δ-closed subsets of Pn × Am are given by the zero sets of collections of differential polynomials in F {y0 , . . . , yn , z1 , . . . , zm } which are homogeneous in y¯. Usually we will consider irreducible Δ-closed sets, that is, those which are defined by prime Δ-ideals. Because differentially closed fields have quantifier elimination, the definable sets over F are simply boolean combinations of closed sets, that is, the constructible sets of the Δ-topology. A constructible set is irreducible if its closure is irreducible. Definition 2.4. A Δ-closed X ⊆ Pn is Δ-complete if the second projection π2 : X ×Y → Y is a closed Δ-map for every quasiprojective Δ-variety, Y . It is not a restriction to consider only irreducible Δ-closed sets. To see this, note that in proving that X × Y → Y is Δ-closed, it is enough to prove that the map is Δ-closed on each irreducible component of X. Proposition 2.5. If X is Δ-complete and Y is a quasiprojective Δ-variety, 1. Suppose f : X → Y is a morphism of Δ-varieties. Then f (X) is Δ-closed in Y and Δ-complete. 2. Any Δ-closed subset of X is Δ-complete. 3. Suppose that Y is Δ-complete. Then X × Y is Δ-complete. Proof. Let f : X → Y ⊆ Pm . Then f × id : X × Pm → Pm × Pm is a continuous map. The diagonal of Pm × Pm is Δ-closed. By virtue of the completeness of X, π2 (gr(f )) is Δ-closed. This is f (X). Now, we get the following commuting diagram, giving the completeness of the image, f (X).

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X ×Y π2

f ×id

359

f (X) × Y

π2

Y For 2), simply note that if Z is and Δ-closed subset of X, then we have the natural injective map Z × Y → X × Y . Further, the π2 projection map clearly factors through this map. So, Z must be Δ-complete. Similarly, for 3), we can simply note that if we have X × Y × Z, then the projection X × Y × Z → Y × Z is closed by the completeness of X. Y × Z → Z is closed by the completeness of Y . The composition of closed maps is closed. 2 When we wish to verify that a differential algebraic variety, X, is complete, we need only show that π :X ×Y →Y is Δ-closed for affine Y . This fact is true for the same reason as in the case of algebraic varieties. Δ-closedness is a local condition, so one should cover Y by finitely many copies of Am for some m and verify the condition on each of the affine pieces. Example 2.6. We continue our elliptic curve Example 2.3. Kolchin [10] proved that Pn (C) = Z





xj δ(xi ) − xi δ(xj )

 i,j=1,...,n

⊂ Pn

is differentially complete. Pong [26] gave an alternate proof of this fact. It follows from Proposition 2.5 part 2 that E is differentially complete, being a Δ-closed subset of P2 (C). 3. Pn is not Δ-complete There are more closed sets in Δ-topology on Pn (U) than in the Zariski topology on Pn (U), so in the differential setting, there are more closed sets in both the image of the projection map (which makes completeness “easier” to achieve) and more closed sets in the domain (making completeness “harder” to achieve for partial differential fields). Specifically, recall that verifying that a given X is Δ-complete involves showing that π2 : X × Y → Y is a closed map for arbitrary Δ-variety Y . So, there are more closed sets to worry about in the domain, but this is offset by the fact that there are more closed sets in the range. Though the Δ-topology is richer than the Δ-topology for ordinary differential fields, Pong’s exposition [26] of Kolchin’s theorem that Pn is not complete holds in this setting. The techniques are model-theoretic, as in [26], but use the model theory of Δ-fields.

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Consider, for some δ ∈ Δ, the closed set in P1 × A1 given by solutions to the equations z(δy)2 + y 4 − 1 = 0,

(1)

2zδ 2 y + δzδy + 4y 3 = 0.

(2)

Note that the projective closure with respect to the y variable, does not contain the point at infinity. So, we can work with the above local equations. Let b ∈ U be a Δ-transcendental over F . Then, we have a solution, in U, to the equation b(δy)2 + y 4 − 1 = 0. Further, we can demand that 4y 3 − 1 = 0. Call this solution a. This means that δa = 0. But, since we chose b to be a Δ-transcendental over F , we know, by quantifier elimination, that if π2 Z is Δ-closed, then it is all of A1 . But π2 Z cannot be all of A1 , since π2 Z cannot contain 0. Thus, π2 Z is not Δ-closed and P1 is not Δ-complete. Since P1 is a Δ-closed subset of Pn for any n, we have the following result of Kolchin. Proposition 3.1. Pn (U) is not Δ-complete. If m > 1, there are proper definable subfields of U, i.e., Kolchin closed Δ-subfields. They are all fields U1 of constants of nonempty commuting linearly independent sets Δ1 of derivations D = UΔ, the vector space generated by Δ. For a careful analysis, see [31]. If the cardinality of Δ1 is m − m1 > 0, the differential field U1 is a model of DCF 0,m1 , and is clearly not Δ1 -complete. It also fails to be Δ-complete. 4. Embedding theorems In this section, we think of the differential algebraic varieties which appear as synonymous with their U rational points where U is a very saturated differentially closed field. We will use the model-theoretic tool of Lascar rank to show that every Δ-complete set can be embedded in A1 . We may identify hypersurfaces of degree d in Pn with points in d+n P d −1 via the following correspondence,

a ¯ = [a1 . . . ad+n ] ∈ P d

d+n d

−1

↔ Ha¯ = Z

d+n d 

ai Mi

i=1

where Mi is the ith monomial of degree d in x0 . . . xn ordered lexicographically. Notice the unusual numbering on the parameter space of ai ’s. This follows Hartshorne [7]; let   N = d+n − 1 for the remainder of the section. We say that a hypersurface is generic d if the associated a ¯ is a tuple of independent differential transcendentals over F . This is equivalent to saying that RU (a/F ) = ω m · N .

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Proposition 4.1. Suppose X ⊂ Pn is a definable set of Lascar rank less than ω m . Then any generic hypersurface does not intersect X. Proof. Let

H=Z

d+n d 

ai Mi

i=1

where a ¯ is a generic point in PN . Now, suppose that x ∈ X ∩H. Then x specifies a proper  subvariety of PN , namely the hyperplane given by ai Mi (x) = 0. This hyperplane is, N −1 of course, isomorphic to P . But, we know that the rank of a generic point in PN −1 is m not ω · N , namely we know RU (a/x) ≤ ω m · (N − 1). Now, using the Lascar inequality, RU (a) ≤ RU (a, x) ≤ RU (a/x) ⊕ RU (x) < ω m · N. So, a ¯ must not have been generic in PN .

2

Proposition 4.2. Let X ⊆ Pn be a definable set whose F -closure is not all of Pn . Let p ∈ Pn − X be generic. Suppose RU (X) ≥ ω m . Then RU (πp (X)) ≥ ω m , where πp is projection from the point to any hyperplane not containing the point. Proof. Take b ∈ X generic (of full RU -rank). Then let c = πp (b). Since we assumed that p ∈ Pn − X, we know that the intersection of the line pb and X is a proper Δ-closed subset of the affine line (note that in what follows, pb denotes the line through points p and b; it is also standard model theoretic notation to write RU (b/cp) or RU (b/c, p) for RU (b/{c} ∪ {p})). Thus, RU (b/c, p) < ω m . But then ω m ≤ RU (b) = RU (b/p) ≤ RU (b, c/p) ≤ RU (b/p) ⊕ RU (c/b, p). Of course, this implies that RU (c) ≥ ω m .

2

Remark 4.3. The previous proposition is true under weaker assumptions on p when X is closed, as the referee suggested. Let p be any point of X. Given any point b on X, the line pb has a canonical embedding as a closed subset of A1 , thought of as pb. Obviously, it is a proper closed subset when p ∈ / X. Thus the rank of its generic type is strictly less than ω m . When projecting X to a hyperplane from p, this means that the rank of the fibers of the maps is strictly bounded by ω m , so the rank of the image must be at least ω m by the Lascar inequality. Proposition 4.4. Let k ∈ N and let X be a Δ-variety with Lascar rank less than ω m (k+1). Then X is isomorphic to a definable subset of P2k+1 .

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Proof. Suppose that X ⊆ Pn . Let p be a generic point of Pn . Let H be any hyperplane not containing p. We claim that projection from p to H, restricted to X, is an injective map. Suppose not. Then there are two points x1 , x2 on X, so that p is on the line joining x1 and x2 . Then RU (p) ≤ RU (p, x1 , x2 ) ≤ RU (p/x1 , x2 ) ⊕ RU (x1 x2 ) < ω m · (2k + 2). This is a contradiction unless n < 2k + 1. Iteratively projecting from a generic point gives the result. 2 Remark 4.5. In the special case that X is an algebraic variety, then this simply says that we can construct a definable isomorphism to some constructible set in P2 dim(X)+1 . From the previous two propositions, we get, Corollary 4.6. Suppose that X is of Lascar rank less than ω m . Then X is definably isomorphic to a definable subset of A1 . Proof. Using Proposition 4.4, we get an embedding of X into P1 . We know that X avoids any generic point of P1 by Proposition 4.1. 2 Remark 4.7. The use of Proposition 4.1 in the above proof is gratuitous, since it is clear that the projection of X is a proper subset of P1 , by simple rank computations. The proposition is less obvious when the variety is embedded in higher projective spaces. Theorem 4.8. Any Δ-complete set is of Lascar rank strictly less than ω m . Proof. Suppose X is complete and of rank larger than ω m . Projection from any generic point gives a Δ-complete, Δ-closed set (by Proposition 2.5) of rank at least ω m (by Proposition 4.2). Iterating the process yields such a set in P1 . The only Δ-closed subset of rank ω m in P1 is all of P1 . This is a contradiction since P1 is not Δ-complete. 2 Corollary 4.9. Every Δ-complete subset of Pn is isomorphic to a Δ-closed subset of A1 . Remark 4.10. More results along the lines of those in [25] computing bounds on ranks of various algebraic geometric constructions on differential varieties are certainly possible, but the above results suffice our purposes here. The astute reader may notice that we are routinely enlarging the field of definition in order to prove various embedding theorems, since the techniques involved operations along the lines of projection from a generic point. This enlargement of the field of definition is a convenient way to phrase the proofs, but is ultimately not necessary. For instance, if projection from a generic point is an injective map on a differential algebraic variety V , then the map must be injective on some open subset of the ambient space (since injectivity is a first-order property). Since our field F is assumed to be Δ-closed, there is some F -rational point in this open set.

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The following two questions may or may not be distinct. For discussion of this see [6] and [31]. Question 4.11. Are there Δ-complete F -closed sets of infinite Lascar rank? The answer is no in the case |Δ| = 1. Question 4.12. Are there Δ-complete F -closed sets whose fields of Δ-rational functions have infinite transcendence degree over F? In the case that |Δ| = 1, this question is equivalent to the previous question. Example 4.13. We continue with the elliptic curve Example 2.3 (reminder: E(C) consists of the C-points of an elliptic curve defined with coefficients in C). We wish to give concrete examples of Proposition 4.1 and Proposition 4.2. These propositions are used to give differential rational embeddings of E(C) (or any differential algebraic group of differential transcendence degree zero) into affine space (Proposition 4.1) and P1 (Proposition 4.2). Proposition 4.1 will give an embedding of E(C) into A2 . Generally, the content of the proposition in the ordinary case is that any closed set of Pn with finite transcendence degree defined over F does not intersect a line with coefficients which are generic over F . To that end, consider the line L determined by αx + βy + z = 0 where α and β are Δ-algebraically independent over F . Then L ∩ E(C) = ∅, so the map φ : E(C) → A2 given by  [x, y, z] →

y x , αx + βy + z αx + βy + z



is a well-defined injective map. The inverse map φ−1 : im(φ) → E is given by (u, v) → [u : v : 1 − αu − βv]. The image φ(E(C)) ⊂ A2 is given by the zero set of the following differential polynomials over F α, β : (1 − αu − βv)v 2 − u3 − a(1 − αu − βv)2 u + b(1 − αu − βv)3 δu − βvδu + δαu2 + δβuv + βuδv δu − αuδv + δαuv + αvδu + δβv 2 . In this specific example, E(C) is actually a differential algebraic group of finite transcendence degree. More specifically, it is a differential algebraic subgroup of an algebraic group. Then we obtain a group operation on the image of E(C) in A2 .

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Recall that we assume that F is differentially closed. Since the set of lines which intersect E(C) is a definable subset of the Grassmannian, GR(1, 2), by the model completeness of DCF 0,m (see [16] for the ordinary case or [17] for the partial case), there is a line defined over F which does not intersect E(C). With more care, one could construct a morphism with all of the pertinent properties of φ which is defined over F . For instance, in the particular case given above, any line of the form x + γy = 0 where γ is not a constant does not intersect E(C). Remark 4.14. A Δ-variety V has Δ-transcendence degree zero if and only if V has Lascar rank less than ω m . More generally, using Proposition 4.1 and a suitable generalization of the diagram and the model completeness mentioned in the previous paragraph: Corollary 4.15. Every differential algebraic variety of Δ-transcendence degree zero defined over F is isomorphic to an affine differential algebraic variety over F . Every differential algebraic group of Δ-transcendence degree zero defined over F is isomorphic to an affine differential algebraic group over F . Next, we will use Proposition 4.4 to give an embedding of E into P1 . Take [α : β : 1] generic in P2 over F . By Proposition 4.4 the morphism, π, obtained by projection (of E) from this point to any hyperplane in P2 is injective. For instance, we might choose the hyperplane given by z = 0. In this case, the map π : E → P1 is given by [x : y : z] → [x − αz : βz − y]. The map π −1 is given by  uδ(v) − vδ(u) − α  vδ(u) − uδ(v) : [u : v] → uv + δ(β) δ(α) β

δ(β) β uv

vδ(u) − uδ(v) . : δ(α)

Any finite transcendence degree differential algebraic variety in P1 must be a proper subvariety. Naturally, the image of E under the above map has this property. Indeed, the second coordinate of the image never vanishes for [x : y : z] ∈ E. To see this, note that y and z do not simultaneously vanish on E; the unique point of E on the line at infinity is [0 : 1 : 0]. On the other hand, in E, yz is never equal to β, since β is a Δ–F -transcendental (remember, E has finite transcendence degree). So, the map ψ : E → A1 is given by [x : y : z] →

x − αz . βz − y

Naturally, this gives another result along the lines of Corollary 4.15:

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Corollary 4.16. Every differential algebraic variety of Δ-transcendence degree zero over F is isomorphic over F to a constructible set in A1 . Every differential algebraic group of Δ-transcendence degree zero defined over F is isomorphic over F to a differential algebraic group embedded in A1 . Question 4.17. Restrict to the ordinary case. Are all projective finite Morley rank differential algebraic varieties complete? In the partial case, this is a special case of each of the following two questions. Are all projective differential transcendence degree zero differential algebraic varieties complete? Are all projective finite transcendence degree differential algebraic varieties complete? Remark 4.18. Since the initial versions of this article, progress on this problem has resulted in a negative answer to the first question asked above. For details, see [30]. Remark 4.19. Given the stark contrast of differential completeness with the classical notion in algebraic geometry, one should not expect theorems from the algebraic category which use the essentially projective nature of complete algebraic varieties to hold for complete differential algebraic varieties. For instance, complete algebraic varieties have so few regular functions that: Proposition 4.20 (Rigidity theorem). Let α : V × W → U be a regular map, let V be complete and V × W geometrically irreducible. Then if there are u0 ∈ U (k), v0 ∈ V (k), and w0 ∈ W (k) so that     α V × {w0 } = {u0 } = α {v0 } × W then α(V × W ) = {u0 }. For a proof, see [19]. Of course, this theorem does not hold for differential algebraic varieties (in this setup, a regular function on a Kolchin-open subset of a differential variety is a differential rational function which is well-defined at the points of the open set). For a simple counterexample, consider the projective closure of the affine differential algebraic variety Z(x −x3 ) ⊂ A1 . The closure, in P1 is given by the zero set of differential homogenization of the differential polynomial, that is V = Z(y(x y − xy  ) − x3 ) ⊆ P1 . V is complete [29]. Notice that the point at infinity, [1 : 0] is not a point on this variety. So, xy is a regular function on V . Now, consider α : V × A1 → A1 given by α([x, y], z) = xy z. The coordinate axes are mapped to zero, but the image of the map is all of A1 .

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5. A valuative criterion for Δ-completeness The following is a proposition of van den Dries [32] mentioned in the introduction, which Pong used in the case of ordinary differential fields to establish a valuative criterion for completeness. We take a similar approach here. Proposition 5.1. T a complete L-theory and φ(v1 . . . vn ) an L-formula without parameters. Then the following are equivalent: 1. There is a positive quantifier free formula ψ such that T proves ∀vφ(v) ↔ ψ(v). 2. For any K, L |= T and f : A → L an embedding of a substructure A of K into L, if a ∈ Am and K |= φ(a) then L |= φ(f (a)). Remark 5.2. Here substructure is taken in the sense of model theory and depends upon the language and theory. Proposition 5.3. Let R ⊇ Q be a Δ-ring. Every proper Δ-ideal I is contained in some prime Δ-ideal. Proof. See [8, see II.5].

2

The next definition is essential for the criterion we give for completeness. Definition 5.4. Let K be a Δ-field.  HK := (A, f, L) : A is a Δ-subring of K, L is a Δ-field,  f : A → L a Δ-homomorphism . Given (A1 , f1 , L1 ), (A2 , f2 , L2 ) ∈ HK . Then f2 extends f1 if A1 ⊂ A2 , L1 ⊆ L2 , and f2 |A1 = f1 . In this case, one could write (A1 , f1 , L1 ) ≤ (A2 , f2 , L2 ). With respect to this ordering, HK has maximal elements. These will be called maximal Δ-homomorphisms of K. Remark 5.5. There are no assumptions (yet) about these homomorphisms being over F . Eventually, we will enforce this condition model-theoretically by changing the formal language. Definition 5.6. A Δ-subring is maximal if it is the domain of a maximal Δ-homomorphism. A Δ-ring is called a local Δ-ring if it is local and the maximal ideal is a Δ-ideal. Proposition 5.7. Let (R, f, L) be maximal in HK . Then, 1. R is a local Δ-ring and ker(f ) is the maximal ideal. 2. x ∈ K − R if and only if m{x} = R{x}.

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Proof. Kernels of Δ-homomorphisms are Δ-ideals. Let x ∈ R, x ∈ / ker(f ). Then we −1 −1 −1 extend f to R(x) by letting x → f (x) . By maximality, x ∈ R. Thus, x is a unit. This establishes the first statement. k  If m{x} = R{x} then 1 = i=1 mi pi (x). Then if x ∈ R, f (1) = f ( mi pi (x)) = 0, a contradiction. For the converse: if m{x} = R{x}, then there is a prime Δ-ideal m which contains m{x}. So, we let K  be the fraction field of R{x}/m . L

K

L

K Further, we have: R{x}

K

L

R

K

L

But, one can see that the maximality condition on R means that x ∈ R.

2

Remark 5.8. We are about to give the valuative criterion for completeness. The varieties in question are given as closed subsets of affine space. This is slightly deceptive, especially if one wishes to think in analogy to classical results of algebraic geometry. In light of the results of Section 3, any complete differential algebraic variety might be equipped with an affine embedding. For the previous results of this section, we considered substructures in the language of differential rings, that is Δ-subrings. So, there was no assumption about the differential homomorphisms fixing F . For the next result, we take T to be Th(F ) in the language of differential rings, {Δ, 0, 1, +, ·} augmented by constant symbols {d}d∈F where F is a differentially closed field (this approach was mentioned in the introduction). The substructures in this augmented language are Δ–F -algebras. Homomorphisms of substructures in this language are differential ring homomorphisms which fix each element of F . The models of T are differentially closed fields which contain F . Theorem 5.9. Let X be a Δ-closed subset of An . Then X is Δ-complete if and only if for any K |= T and any R, a maximal Δ-subring of K containing F , we have that every K-rational point of X is an R-rational point of X.

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Proof. Suppose the valuative criterion holds for X. Then, to show that X is Δ-complete, it suffices to show that for any Δ-closed set Z ⊆ X ×An , π2 (Z) is Δ-closed. Given K and f with K, L |= DCF 0,m , let f : A → L be a Δ-homomorphism where A is a substructure of K. We let φ(y) be the formula saying y ∈ π2 Z. We will show that if a ∈ An and K |= φ(a) then L |= φ(f (a)). So, we assume there is an x ∈ X(K) with (x, a) ∈ Z. Now, extend f to f˜ : R → L a maximal Δ-homomorphism. One can always assume that L |= DCF 0,m , since if this does not hold, we simply take the Δ-closure. At this point, we have F ⊆ A ⊆ R. By the assumption, we know already that x ∈ R.   L |= f˜(x), f˜(a) ∈ Z ∧ f˜(x) ∈ X. So, L |= f˜(a) ∈ π2 Z. But, then L |= f (a) ∈ π2 Z, since f˜(a) = f (a). Now, using van Den Dries’ condition, Proposition 5.1, we can see that π2 Z is Δ-closed. Now, we suppose that the valuative criterion does not hold of X. So, we have some f : R → L a maximal Δ-homomorphism of K, with R ⊇ F . There is some point x ∈ X(K), so that x ∈ / R. Then for one of the elements xi in the tuple x, we know, by Proposition 5.7 that 1 ∈ m{xi }. Then, we know that there are mj ∈ m and tj ∈ R{y} so that 

mj tj (x) = 1.

j

We let m := (m1 . . . mk ). So, we let Z be the differential algebraic variety given by the conditions 

zj tj (yi ) − 1 = 0 and y ∈ X.

j

Then we take L to be the Δ closure of the Δ-field R/m. If we let g be the quotient map then g|F is an embedding. Then we have that K |= ∃yφ(y, m) since x is a witness. But, we see that L cannot have a witness to satisfy this formula, m ∈ ker(f ). Then again, by Proposition 5.1, we see that π2 Z is not Δ-closed. 2 One can rephrase the criterion for an affine Δ-closed subset X, a fact noticed by Pong, in the ordinary case. For any K |= DCF 0,m , let R be a maximal Δ-subring of K which contains F . Let A = K{y1 , . . . , yn }/I(X). Suppose we are given a commutative diagram, Spec K

Spec A

Spec R

Spec F

then we have the diagonal morphism,

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Spec K

Spec A

Spec R

Spec F

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6. Δ-complete varieties In this section, we will use some commutative algebra along with the valuative criterion given above to give examples of Δ-complete sets. Let y¯ = (y1 , . . . , yn ) and consider a system of linear differential equations of the form {δi y¯ = Ai y¯}m i=1 where Ai ∈ gln (F ). The system is integrable if δ i A j − δ j A i = Ai A j − A j A i for 1 ≤ i, j ≤ m. In the case n = 1, one can quite clearly see that the conditions are a special case of the notion of coherence present in the axioms for differentially closed fields: Example 6.1. Let Δ = {δ1 , δ2 }. Let n = 1. Then we are considering the system of equations given by the zero set of two differential polynomials, f1 , f2 : f1 (y) = δ1 y − a1 y = 0 f2 (y) = δ2 y − a2 y = 0. In this case, the system is coherent if δ2 (f1 (y)) − δ1 (f2 (y)) ∈ I(f1 , f2 ). But     δ2 f1 (y) − δ1 f2 (y) = δ2 (δ1 y − a1 y) − δ1 (δ2 y − a2 y) = δ2 δ1 (y) − δ2 (a1 )y − a1 δ2 (y) − δ1 δ2 (y) + δ1 (a2 )y − a2 δ1 (y) taking into account the relations of I(f1 , f2 ), we see:     δ2 f1 (y) − δ1 f2 (y) =mod(I(f1 ,f2 )) δ2 (a1 )y − δ1 (a2 )y. Clearly, the expression on the right hand side is zero only if δ2 (a1 ) − δ1 (a2 ) = 0. The integrability conditions given here are standard hypotheses in differential Galois theory [3, section two]. If the conditions are not satisfied, then the only point in the solution set of the system is zero. Since we are looking for examples of complete Δ-varieties, we want to rule out the scenario that the variety is simply a point. Generally speaking, this example is indicative of the complications with specifying classes of partial differential varieties:

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The class of examples we wish to consider in this section will be both more and less general than the preceding example. We consider systems of equations in one variable which are nonlinear. 1 Theorem 6.2. Let V be the Δ-closure of Z({δi y − Pi (y)}m i=1 ) in P , where Pi (y) ∈ F [y], the collection {δi y − Pi (y)}m i=1 is coherent as a collection of differential polynomials, and at least one of the Pi is of degree greater than one. Then V is Δ-complete.

Remark 6.3. Note that since one of the polynomials Pi is at least quadratic, one can show that the system is projectively closed, that is the point at infinity is not in the projective closure of this affine differential algebraic variety. Before beginning the proof, we give a brief instance of the theorem and also point out the restriction that the coherence conditions impose – the reader unfamiliar with this notion might consult [27,11]. Technically, the coherence conditions are not required for the formal truth of the theorem; if the system is not coherent, the set of solutions is either empty or finite. Thanks to the referee for pointing out that “integrability” or coherence conditions are necessary for the theorem to be interesting; also, the specific example in this remark was also suggested by the referee. One might call the family of equations in Theorem 6.2 a family of generalized Ricatti equations in m variables. Let us assume that m = 2 and Δ = {δ1 , δ2 }. Consider the following family of equations: δ1 y = a2 y 2 + a1 y + a0 2

δ2 y = b2 y + b1 y + b0 .

(3) (4)

In this example, coherence implies that the following equations must be satisfied: δ2 a2 − δ1 b2 + a2 b1 − a1 b2 = 0

(5)

δ2 a1 − δ1 b1 + 2(a2 b0 − a0 b2 ) = 0

(6)

δ2 a0 − δ1 b0 + a1 b0 − a0 b1 = 0.

(7)

Viewing the coefficients themselves as variables in a differential field, and taking the leaders of the equations to be δ2 a2 , δ2 a1 , δ2 a0 (there are a variety of rankings which would ensure this), one can see that the system is coherent and autoreduced for any choices of b0 , b1 , b2 ; thus by the axioms for differentially closed fields, there are choices of the ai which guarantee coherence for any choice of b0 , b1 , b2 . Remark 6.4. We note that the following proof was much more complicated in previous versions of this paper, where a different strategy was followed. The referee suggested the following proof, which works via reduction to the ordinary case. Proof. One should homogenize δi y − Pi y in P1 to calculate the Δ-closure. If deg Pi = di > 1 then Hi (y, y1 ) = y1di (δi ( yy1 ) − Pi ( yy1 )) is homogeneous. One can easily see (by

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examining the leading monomial of y1di Pi ( yy1 )) that [1, 0] ∈ / Z(Hi (y, y1 )). Thus, the set V in affine space is equal to its projective closure. Now, we can use the valuative criteria to establish the completeness of the variety. Let x ∈ V (K) and R a maximal Δ-subring containing F . It is enough to show that mK {x} = R{x}. Since δi x = Pi (x), m{x} = m[x] R{x} = R[x]. Fix some i so that deg Pi ≥ 2. Now, extend (R, f, L) to (Ri , fi , Li ), a maximal δi − F homomorphism which extends f . Then F ⊂ R ⊂ Ri ⊂ K and Ri is a maximal δi -ring containing R and mi ∩ R = m. By the fact that x is monic over Ri , we know that Ri {x} = Ri [x] and mi {x} = mi [x]. Now, by [26, proof of Theorem 6.3] we see that 1∈ / mi [x] and thus 1 ∈ / m[x]. 2 Remark 6.5. Beyond order 1 the techniques as shown above are not as easy to apply. For techniques in that situation, at least in the case of linear ordinary differential varieties, see [29]. One can combine the above techniques of that paper with the above techniques to give a wider class of complete partial differential varieties. Acknowledgments I would like to thank Dave Marker and William Simmons for many helpful discussions on this topic. I would also like to thank the participants of the Kolchin seminar and Alexey Ovchinnikov in particular for useful discussions and encouragement. The anonymous referee provided many useful comments and suggestions which affected the exposition as well as some of the technical aspects of this paper. In particular, the referee suggested the specific series of examples regarding the constant points of the elliptic curve. I should note that Phyllis Cassidy also deserves credit for this as she suggested this example several years prior during a visit to the Kolchin seminar. References [1] Lenore Blum, Generalized algebraic structures: a model theoretical approach, PhD thesis, MIT, 1968. [2] Peter Blum, Complete models of differential fields, Trans. Amer. Math. Soc. 137 (1969) 309–325. [3] Phyllis J. Cassidy, Michael F. Singer, Galois theory of parameterized differential equations and linear differential algebraic groups, in: Differential Equations and Quantum Groups, in: IRMA Lectures in Mathematics and Theoretical Physics, vol. 9, EMS Publishing House, 2006, pp. 113–157. [4] Phyllis J. Cassidy, Michael F. Singer, A Jordan–Hölder theorem for differential algebraic groups, J. Algebra 328 (2011) 190–217. [5] James Freitag, Generics in differential fields, http://www.math.uic.edu/~freitag/GenericPoints.pdf, 2014, in preparation. [6] James Freitag, Indecomposability in partial differential fields, under review, J. Pure Appl. Algebra (2011), http://arxiv.org/abs/1106.0695.

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[7] Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, 1977. [8] Irving Kaplansky, An Introduction to Differential Algebra, Actualités scientifiques et industrielles, Hermann, 1957. [9] William F. Keigher, Differential schemes and premodels of differential fields, J. Algebra 79 (1) (1982) 37–50. [10] Ellis Kolchin, Differential equations in a projective space and linear dependence over a projective variety, in: Contributions to Analysis (A collection of papers dedicated to Lipman Bers), 1974, pp. 195–214. [11] Ellis R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1976. [12] Jerald J. Kovacic, Differential schemes, in: Differential Algebra and Related Topics, World Science Publishing, 2002, pp. 71–94. [13] Jerald J. Kovacic, Global sections of diffspec, J. Pure Appl. Algebra 171 (2–3) (2002) 265–288. [14] Jerald J. Kovacic, Differential schemes, in: Kolchin Seminar in Differential Algebra, 2005. [15] David Marker, Model Theory: An Introduction, Second edition, Graduate Texts in Mathematics, vol. 217, Springer, 2002. [16] David Marker, Margit Messmer, Anand Pillay, Model Theory of Fields, A.K. Peters/CRC Press, 2005. [17] Tracy McGrail, The model theory of differential fields with finitely many commuting derivations, J. Symbolic Logic 65 (2) (2000) 885–913. [18] Ronald Bustamante Medina, Rank and dimension in difference-differential fields, preprint. [19] James Milne, Abelian varieties, Lecture notes, www.jmilne.org, 2008. [20] Rahim Moosa, Anand Pillay, Thomas Scanlon, Differential arcs and regular types in differential fields, J. Reine Angew. Math. 620 (2008) 35–54. [21] Sally Morrison, Extensions of differential places, Amer. J. Math. 100 (2) (1978) 245–261. [22] Masayoshi Nagata, Existence theorems for nonprojective complete algebraic varieties, Illinois J. Math. 2 (1958) 490–498. [23] Anand Pillay, Geometric Stability Theory, Oxford University Press, 1996. [24] Anand Pillay, Remarks on algebraic D-varieties and the model theory of differential fields, in: Logic in Tehran, 2003. [25] Wai-Yan Pong, Ordinal dimensions and completeness in differentially closed fields, PhD thesis, 1999. [26] Wai-Yan Pong, Complete sets in differentially closed fields, J. Algebra 224 (2000) 454–466. [27] Azriel Rosenfeld, Specializations in differential algebra, Trans. Amer. Math. Soc. (1959) 394–407. [28] I.R. Shafarevich, Schemes and Complex Manifolds, Basic Algebraic Geometry, Springer-Verlag, 1994. [29] William Simmons, Differential completeness and finite transcendence degree differential varieties, 2014, in preparation. [30] William D. Simmons, Completeness of finite-rank differential varieties, PhD thesis, University of Illinois at Chicago, 2013. [31] Sonat Suer, Model theory of differentially closed fields with several commuting derivations, PhD thesis, University of Illinois at Urbana–Champaign, 2007. [32] Lou van den Dries, Some applications of a model theoretic fact to (semi-) algebraic geometry, Nederl. Akad. Wetensh. Indag. Math. 44 (4) (1982) 397–401.