MULTIVARIABLE DIMENSION POLYNOMIALS AND NEW INVARIANTS OF DIFFERENTIAL FIELD EXTENSIONS

We introduce a special type of reduction in the ring of differential polynomials and develop the appropriate technique of characteristic sets that allows to generalize the classical Kolchin’s theorem on differential dimension polynomial and find new differential birational invariants of a finitely generated differential field extension. 2000 Mathematics Subject Classification. 12H05, 12H20, 13N15.


Introduction.
The role of Hilbert polynomials in commutative algebra and algebraic geometry is well known.A similar role in differential algebra is played by differential dimension polynomials.The notion of a differential dimension polynomial was introduced by Kolchin in [6], but the problems and ideas that had led to this concept have essentially more long history.Actually, the differential dimension polynomial describes in exact terms the freedom degree of a dynamic system as well as the number of arbitrary constants in the general solution of a system of algebraic differential equations.The first attempts of such a description were made in the 19th century by Jacobi [3] who estimated the number of algebraically independent constants in the general solution of a system of ordinary linear differential equations.Later on, Jacobi's results were extended to some nonlinear systems, but in the general case the problem of such estimation (known as the problem of Jacobi's bound) remains open.
Differential algebra as a separate area of mathematics is largely due to its founder Ritt  and Kolchin (1916Kolchin ( -1991)).In 1964 Kolchin proved his famous theorem on differential dimension polynomial (see Theorem 2.1 below) that lies in the foundation of the theory of differential dimension.At the International Congress of Mathematicians in Moscow (1966) Kolchin formulated the main problems and outlined the most perspective directions of research connected with the differential dimension polynomial.Later on the results obtained in this area were included into his famous monograph [7] that hitherto remains the most fundamental work on differential algebra.
Discussing the history of creation of the differential dimension theory, one should note that in 1953 Einstein [2] introduced a concept of strength of a system of differential equations as a certain function of integer argument associated with the system.In 1980 Mikhalëv and Pankrat'ev [12] showed that this function actually coincides with the appropriate differential dimension polynomial and found the strength of some well-known systems of partial differential equations using methods of differential algebra.
The intensive study of Kolchin's differential dimension polynomials began at the end of the sixties with the series of works by Johnson [4,5,15] who developed the technique of dimension polynomials for differential modules and applied it to the study of some classical problems of differential algebra.In particular, he characterized the Krull dimension of finitely generated differential algebras, developed the theory of local differential algebras, and proved a special case of Janet conjecture.A number of interesting properties and applications of differential dimension polynomials were found by Kondrat'eva, Levin, Mikhalëv, Pankrat'ev, Sit, and some other mathematicians (see [9,10,11,12,13,14]).One of the most important directions of this study was the search for new differential birational invariants connected with the differential dimension polynomials.Here we should mention the results of Sit [13] who showed that the set of all differential dimension polynomials is well ordered with respect to some natural ordering and introduced the notion of the minimal differential dimension polynomial associated with a differential field extension.
In this paper, we introduce a special type of reduction in a ring of differential polynomials over a differential field of zero characteristic whose basic set is represented as a disjoint union of its subsets.Using the idea of the Gröbner basis method introduced in [1], we develop the appropriate technique of characteristic sets that allows to prove the existence and outline a method of computation of multivariable dimension polynomials associated with a finitely generated differential field extension.In particular, we obtain a generalization of the Kolchin's theorem and find new differential birational invariants.

Preliminaries.
Throughout the paper Z, N, and Q denote the sets of all integers, all nonnegative integers, and all rational numbers, respectively.By a ring we always mean an associative ring with a unit.Every ring homomorphism is unitary (maps unit onto unit), every subring of a ring contains the unit of the ring, and every algebra over a commutative ring is unitary.Unless otherwise indicated, every field is supposed to have zero characteristic.
A differential ring is a commutative ring R considered together with a finite set ∆ of mutually commuting derivations of the ring R into itself.The set ∆ is called a basic set of the differential ring R that is also called a Let R and S be two differential rings with the same basic set ∆ = {δ 1 ,...,δ m }, so that elements of the set ∆ act on each of the rings as mutually commuting derivations.A ring homomorphism φ : In what follows, K denotes a differential field whose basic set of derivation operators ∆ is a union of p disjoint finite sets (p ≥ 1) : ∆ = ∆ 1 ••• ∆ p , where ∆ i = {δ i1 ,...,δ im i } (i = 1,...,p and m 1 ,...,m p are positive integers whose sum is equal to m, the number of elements of the set ∆).In other words, we fix a partition of the basic set ∆. Let Θ i be the free commutative semigroup generated by the elements of the set ∆ i (i = 1,...,p) and Θ the free commutative semigroup generated by the whole set ∆.For any element θ = δ ..,p) and ord θ = p i=1 ord i θ will be called the order of θ with respect to ∆ i and the order of θ, respectively.As usual, if θ, θ ∈ Θ, we say that θ divides θ if θ = θ θ for some element θ ∈ Θ.By the least common multiple of the elements . This element will be denoted by lcm(θ 1 ,...,θ q ).
A subfield K 0 of the ∆-field K is said to be a differential (or ∆-) subfield of K if δ(K 0 ) ⊆ K 0 for any δ ∈ ∆.If K 0 is a ∆-subfield of the ∆-field K and Σ ⊆ K, then the intersection of all ∆-subfields of K containing K 0 and Σ is the unique ∆-subfield of K containing K 0 and Σ and contained in every ∆-subfield of K containing K 0 and Σ.It is denoted by K 0 Σ .If K = K 0 Σ and the set Σ is finite, Σ = {η 1 ,...,η n }, then K is said to be a finitely generated ∆-extension of K 0 with the set of ∆-generators {η 1 ,...,η n }.In this case we write K = K 0 η 1 ,...,η n .It is easy to see that the field K 0 η 1 ,...,η n coincides with the field Now we can formulate the Kolchin's theorem on differential dimension polynomial (see [7,Chapter 2,Theorem 6]).As usual, If Y = {y 1 ,...,y n } is a finite set of symbols, then one can consider the countable set of symbols in the set of indeterminates ΘY over the differential field K.This polynomial ring is naturally viewed as a ∆-ring where δ(θy j ) = (δθ)y j (δ ∈ ∆, θ ∈ Θ, 1 ≤ j ≤ n) and the elements of ∆ act on the coefficients of the polynomials of R as they act in the field K.The ring R is called the ring of differential (or ∆-)polynomials in the set of differential (∆-)indeterminates y 1 ,...,y n over the ∆-field K.This ring is denoted by K{y 1 ,...,y n } and its elements are called differential (or ∆-) polynomials.
The set of all terms ΘY will be considered together with p orderings that correspond to the orderings of the semigroup Θ and that are denoted by the same symbols < 1 ,...,< p .These orderings of ΘY are defined as follows: By the ith order of a term u = θy j we mean the number ord i u = ord i θ.The number ord u = ord θ is called the order of the term u.
We say that a term u = θy i is divisible by a term v = θ y j (or u is a multiple of v) and write v | u, if i = j and θ | θ.For any terms u 1 = θ 1 y j ,...,u q = θ q y j containing the same ∆-indeterminate y j (1 ≤ j ≤ n), the term lcm(θ 1 ,...,θ q )y j is called the least common multiple of u 1 ,...,u q , it is denoted by lcm(u 1 ,...,u q ).
If A ∈ K{y 1 ,...,y n }, A ∉ K, and 1 ≤ i ≤ p, then the highest with respect to the ordering < i term that appears in A is called the i-leader of the ∆-polynomial A. It is denoted by A ), then I d is called the leading coefficient of the ∆-polynomial A and the partial derivative The leading coefficient and the separant of a ∆-polynomial A are denoted by I A and S A , respectively.
Definition 2.2.Let A and B be two ∆-polynomials from K{y 1 ,...,y n }.We say that A has a lower rank than B and write r kA < r kB if either A ∈ K, B ∉ K, or the vector u

.,ord p u (p) B
with respect to the lexicographic order (where u (1) A and u (1) B are compared with respect to < 1 and all other coordinates of the vectors are compared with respect to the natural order on N).If the two vectors are equal (or A ∈ K and B ∈ K) we say that the ∆-polynomials A and B are of the same rank and write r kA = r kB.
Let K be a ∆-field and G = K η 1 ,...,η n a finitely generated ∆-extension of K with a set of generators η = {η 1 ,...,η n }.Then there exists a natural ∆-homomorphism Φ η from the ring of ∆-polynomials K{y 1 ,...,y n } to G such that Φ η (a) = a for any a ∈ K and Φ η (y j ) = η j for j = 1,...,n.If A ∈ K{y 1 ,...,y n }, then the element Φ η (A) is called the value of the ∆-polynomial A at η, it is denoted by A(η).Obviously, the kernel P of the mapping Φ η is a prime ∆-ideal of the ring K{y 1 ,...,y n }.This ideal is called the defining ideal of η over K or the defining ideal of the ∆-field extension G = K η 1 ,...,η n .It is easy to see that if the quotient field Q of the factor ring R = K{y 1 ,...,y n }/P is considered as a ∆-field (where δ(u/v) = (vδ(u) − uδ(v))/v 2 for any u, v ∈ R), then Q is naturally ∆-isomorphic to the field G. (The appropriate ∆isomorphism is identical on K and maps the images of the ∆-indeterminates y 1 ,...,y n in the factor ring R onto the elements η 1 ,...,η n , respectively.)
It is clear that any polynomial with integer coefficients is numerical.As an example of a numerical polynomial with noninteger coefficients one can consider a polynomial of the form where m 1 ,...,m p ∈ N (p ∈ N, p ≥ 1).If f (t 1 ,...,t p ) is a numerical polynomial, then deg f and deg t i f (1 ≤ i ≤ p) will denote the total degree of f and the degree of f relative to the variable t i , respectively.The following theorem proved in [8] gives the "canonical" representation of a numerical polynomial in several variables.Theorem 3.1.Let f (t 1 ,...,t p ) be a numerical polynomial in p variables t 1 ,...,t p , and let deg t i f = m i (m 1 ,...,m p ∈ N).Then the polynomial f (t 1 ,...,t p ) can be represented in the form with integer coefficients ) that are uniquely defined by the numerical polynomial.
In the rest of the section we deal with subsets of N m where the positive integer m is represented as a sum of p nonnegative integers m 1 ,...,m p (p ∈ N,p ≥ 1).In other words, we fix a partition (m 1 ,...,m p ) of the number m.
If Ꮽ ⊆ N m and r 1 ,...,r p ∈ N, then Ꮽ(r 1 ,...,r p ) will denote the set Furthermore, V Ꮽ will denote the set of all m-tuples v = (v 1 ,...,v m ) ∈ N m that are not greater than or equal to any m-tuple from Ꮽ with respect to the product order on N m .(Recall that the product order on the set The following two theorems proved in [8] generalize the well-known Kolchin's result on the numerical polynomials associated with subsets of N (see [7, Chapter 0, Lemma 17]) and give the explicit formula for the numerical polynomials in p variables associated with a finite subset of N m .Theorem 3.2.Let Ꮽ be a subset of N m where m = m 1 +•••+m p for some nonnegative integers m 1 ,...,m p (p ≥ 1).Then there exists a numerical polynomial ω Ꮽ (t 1 ,...,t p ) with the following properties: It is clear that if Ꮽ ⊆ N m and Ꮽ is the set of all minimal elements of the set Ꮽ with respect to the product order on N m , then the set Ꮽ is finite and ω Ꮽ (t 1 ,...,t p ) = ω Ꮽ (t 1 ,...,t p ).Thus, Theorem 3.3 gives an algorithm that allows to find a numerical polynomial associated with any subset of N m (and with a given partition of m): one should first find the set of all minimal points of the subset and then apply Theorem 3.3.

Reduction in the ring of differential polynomials.
In what follows we keep the notation and conventions of Section 2. In particular, K{y 1 ,...,y n } denotes the ring of ∆-polynomials over a differential field K whose basic set ∆ is a union of p disjoint sets: A (θ ∈ Θ,θ ≠ 1) such that ord i (θu A , then either ord j u

A B < deg u (1)
A A.
A ∆-polynomial B is said to be reduced with respect to a set of ∆-polynomials Σ ⊆ K{y 1 ,...,y n } if B is reduced with respect to every element of Σ. Definition 4.2.A set of ∆-polynomials Σ ⊆ K{y 1 ,...,y n } is called autoreduced if Σ K = ∅ and every element of Σ is reduced with respect to any other element of this set.
The proof of the following lemma can be found in [7, Chapter 0, Section 17].Lemma 4.3.Let N n = {1,...,n} and let A be an infinite subset of N m ×N n (m, n ∈ N, n ≥ 1).Then there exists an infinite sequence of elements of A, strictly increasing relative to the product order, in which every element has the same projection on N n .This result implies the following statement that will be used below.Lemma 4.4.Let S be an infinite set of terms in the ring K{y 1 ,...,y n }.Then there exists an index j (1 ≤ j ≤ n) and an infinite sequence of terms θ 1 y j ,θ 2 y j ,...,θ k y j ,... ∈ S such that θ k | θ k+1 for all k = 1, 2,.... Theorem 4.5.Every autoreduced set of ∆-polynomials is finite.
Proof.Suppose that Σ is an infinite autoreduced subset of K{y 1 ,...,y n }.Then Σ contains an infinite set Σ such that all ∆-polynomials from Σ have different 1leaders.Indeed, if it is not so, then there exists an infinite set Σ 1 ⊆ Σ such that all ∆-polynomials from Σ 1 have the same 1-leader u.By Lemma 4.3, the infinite set .. ∈ Σ 1 and ≤ P denotes the product order on N p−1 ).Since the sequence {deg u A i | i = 1, 2,...} cannot strictly decrease, there exists two indices i and j such that i < j and deg u A i ≤ deg u A j .We obtain that A j is reduced with respect to A i that contradicts the fact that Σ is an autoreduced set.
Below, the elements of an autoreduced set will be always arranged in order of increasing rank.(Therefore, if we consider an autoreduced set of ∆-polynomials Σ = {A 1 ,...,A r }, then r kA The following two theorems can be proven precisely in the same way as their classical analogs (see [7, Chapter 1, Corollary to Lemma 6 and Proposition 3, page 78]).Theorem 4.6.Let Σ = {A 1 ,...,A r } be an autoreduced set in the ring K{y 1 ,...,y n } and let B be a ∆-polynomial.Then there exist a ∆-polynomial B 0 and nonnegative integers p i ,q i (1 ≤ i ≤ r ) such that B 0 is reduced with respect to Σ, r kB 0 ≤ r kB, and Definition 4.7.Let Σ = {A 1 ,...,A r } and Σ = {B 1 ,...,B s } be two autoreduced sets in the ring of differential polynomials K{y 1 ,...,y n }.An autoreduced set Σ is said to have lower rank than Σ if one of the following two cases holds: (1) there exists k ∈ N such that k ≤ min{r ,s}, r kA i = r kB i for i = 1,...,k − 1 and r kA k < r kB k ; (2) r > s and r kA i = r kB i for i = 1,...,s.If r = s and r kA i = r kB i for i = 1,...,r , then Σ is said to have the same rank as Σ .
Theorem 4.8.In every nonempty family of autoreduced sets of differential polynomials there exists an autoreduced set of lowest rank.
Let J be an ideal of the ring K{y 1 ,...,y n }.Since the family of all autoreduced subsets of J is not empty (e.g., it contains the empty set), Theorem 4.8 shows that the ideal J contains an autoreduced subset of lowest rank.Definition 4.9.Let J be an ideal of the ring of differential polynomials K{y 1 ,..., y n }.Then an autoreduced subset of J of lowest rank is called a characteristic set of the ideal J. Proof.Suppose that B ≠ 0. Then B and elements of Σ whose rank is lower than the rank of B form an autoreduced set Σ .It is easy to see that Σ has a lower rank than Σ that contradicts the fact that Σ is a characteristic set of the ideal J. Proof.First of all, we show that no nonzero element of J is reduced with respect to f .Let 0 ≠ h ∈ J, and let k be the smallest positive integer such that h can be written as for some pairwise distinct elements θ 1 ,...,θ k ∈ Θ and some g 1 ,...,g k ∈ R. In what follows, we suppose that k > 1 (clearly, an element of the form gθf (g ∈ R, θ ∈ Θ) is not reduced with respect to f ) and Since θ k f is linear with respect to θ k u (1) f , one can write each ∆-polynomial g j (1 ≤ j ≤ k−1) as g j = g j +g j (θ k f ), where g j ,g j ∈ R and g j does not contain θ k u (1) f .Then Since gθ k f contains θ k u (1) f and none of g j θ j f (1 ≤ j ≤ k − 1) contains this term, the ∆-polynomial h contains θ k u (1) f and r kf ≤ r kh.Similarly, if θ j i is the maximal element of the set {θ 1 ,...,θ k } relative to the order < i on Θ (2 ≤ i ≤ p), then h contains h .It follows that h is reduced with respect to f and f is an element of the lowest rank in J. Therefore, if Σ = {h 1 ,...,h l } is a characteristic set of J, then r kf = r kh 1 and l = 1, whence {f } is also a characteristic set of the ideal J.

Multivariable differential dimension polynomials and their invariants. Now
we can prove the main theorem on multivariable differential dimension polynomial that generalizes the classical Kolchin's result (see Theorem 2.1).

.,js
for i = 1,...,p, and for any l = 1,...,s,ord k θ > r k − c (k) j 1 ,...,js + a kj l − b kj l if and only if k is equal to one of the numbers k l1 ,...,k lq l } (q 1 ,...,q s are some positive integers from the set {1,...,p} and Thus, it is sufficient to show that Card W (j ≤ m i (i = 1,...,p).But this is almost evident: as in the process of evaluation of Card V j;k 1 ,...,kq (r 1 ,...,r p ) (when we used Theorem 3.2(iii) to obtain formula (5.3)), we see that Card W (j 1 ,...,j s ; k 11 ,...,k sqs ; r 1 ,...,r p ) is a product of terms of the form r k +m k −c (k) j 1 ,...,js −S k m k (such a term corresponds to an integer k such that 1 ≤ k ≤ p and k ≠ k iν for any i = 1,...,s, ν = 1,...,q s ; the number S k is defined as The appropriate numerical polynomial ψ j 1 ,...,js k 11 ,...,ksq s (t 1 ,...,t p ) is a product of p "elementary" numerical polynomials f 1 ,...,f p where f . Since the degree of such a product with respect to any variable t i (1 ≤ i ≤ p) does not exceed m i , this completes the proof of the theorem.Definition 5.2.Numerical polynomial Φ η (t 1 ,...,t p ), whose existence is established by Theorem 5.1, is called a dimension polynomial of the differential field extension G = K η 1 ,...,η n associated with the given system of ∆-generators η = {η 1 ,..., η n } and with the given partition of the basic set of derivation operators ∆ into p disjoint subsets ∆ 1 ,...,∆ p .
The following example shows that a dimension polynomial of a finitely generated differential field extension associated with some partition of the basic set of derivation operators can carry more differential birational invariants of the extension than the classical Kolchin differential dimension polynomial.

Theorem 3 . 3 .
The polynomial ω Ꮽ (t 1 ,...,t p ) whose existence is stated by Theorem 3.2 is called the dimension polynomial of the set Ꮽ ⊆ N m associated with the partition (m 1 ,...,m p ) of m.If p = 1, the polynomial ω Ꮽ is called the Kolchin polynomial of the set Ꮽ. Let Ꮽ = {a 1 ,...,a n } be a finite subset of N m and (m 1 ,...,m p ) (p ≥ 1) a partition of m.Let a i = (a i1 ,...,a im ) (1 ≤ i ≤ n) and for any l ∈ N, 0 ≤ l ≤ n, let Γ (l, n) denote the set of all l-element subsets of the set N n = {1,...,n}.Furthermore, for any σ

Definition 4 . 1 .
Let A, B ∈ K{y 1 ,...,y n } and A ∉ K.The ∆-polynomial B is said to be reduced with respect to A if the following two conditions hold:(i) B does not contain any term θu(1)

Theorem 4 . 10 .
Let Σ = {A 1 ,...,A d } be a characteristic set of an ideal J of the ring of ∆-polynomials R = K{y 1 ,...,y n }.Then an element B ∈ R is reduced with respect to the set Σ if and only if B = 0.

Theorem 4 . 11 .
Let J be a cyclic differential ideal of the ring of ∆-polynomials R = K{y 1 ,...,y n } generated by a linear ∆-polynomial f .Then {f } is a characteristic set of the ∆-ideal J = [f ].