MULTIVARIABLE DIMENSION POLYNOMIALS AND NEW INVARIANTS OF DIFFERENTIAL FIELD EXTENSIONS

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MULTIVARIABLE DIMENSION POLYNOMIALS AND NEW INVARIANTS OF DIFFERENTIAL FIELD EXTENSIONS

ALEXANDER B. LEVIN (Received 16 February 2001)

Abstract.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.

1. Introduction. The role of Hilbert polynomials in commutative algebra and alge- braic geometry is well known. A similar role in differential algebra is played by differ- ential 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 num- ber 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 (1893–1951) and Kolchin (1916–1991). In 1964 Kolchin proved his famous theo- rem on differential dimension polynomial (seeTheorem 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 differen- tial 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 character- ized 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 poly- nomials 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 poly- nomials 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 intro- duced 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 par- ticular, we obtain a generalization of the Kolchin’s theorem and find new differential birational invariants.

2. Preliminaries. Throughout the paperZ,N, andQdenote 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.

Adifferential ringis a commutative ringRconsidered together with a finite set∆ of mutually commuting derivations of the ringRinto itself. The set∆is called abasic setof the differential ringRthat is also called a∆-ring. A subring (ideal)R0of a∆-ring Ris called a differential or∆-subring ofR(respectively, differential or∆-ideal ofR) if R0is closed with respect to the action of any operatorδ∈∆. If a differential (∆-)ring is a field, it is called a differential (or∆-)field.

LetRandSbe two differential rings with the same basic set∆= {δ1, . . . , δm}, so that elements of the set∆act on each of the rings as mutually commuting derivations. A ring homomorphismφ:R→Sis called adifferentialor∆-homomorphismifφ(δa)= δφ(a)for anyδ∈∆,a∈R.

In what follows,Kdenotes a differential field whose basic set of derivation oper- ators∆ is a union ofp disjoint finite sets (p≥1):∆=∆1

···

∆p, where∆i= {δi1, . . . , δim_i}(i=1, . . . , pand m1, . . . , mp 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 Θi be the free commutative semigroup generated by the ele- ments of the set∆i (i=1, . . . , p) andΘthe free commutative semigroup generated

by the whole set∆. For any elementθ=δ^k₁₁¹¹···δ^k_1m^1m₁¹δ^k₂₁²¹···δ^kpm^pmp_p ∈Θ, the numbers ordiθ=m_i

j=1kij (i=1, . . . , p) and ordθ =p

i=1ordiθ will be called the order of θ with respect to∆iand the order ofθ, respectively. As usual, ifθ, θ∈Θ, we say that θdividesθ ifθ=θθfor some elementθ∈Θ. By the least common multiple of the elementsθ1=p

i=1

m_i

j=1δ^k_ij^ij1, . . . , θq=p i=1

m_i

j=1δ^k_ij^ijq∈Θwe mean the element θ=p

i=1

m_i

j=1δ^k_ij^ij, wherekij =max{kijl|1≤l≤q} (1≤i≤p,1≤j ≤mi). This element will be denoted by lcm(θ1, . . . , θq).

For anyr1, . . . , rp,r∈N, the sets{θ∈Θ|ordiθ≤rifori=1, . . . , p}and{θ∈Θ| ordθ≤r}will be denoted byΘ(r1, . . . , rp)andΘ(r ), respectively. Furthermore, we set T ξ= {θ(ξ)|θ∈T}for anyξ∈K,T⊆Θ.

Below we will considerporderings<1, . . . , <ponΘdefined as follows:

θ=δ^k₁₁¹¹···δ^k_1m^1m₁¹δ^k₂₁²¹···δ^kpm^pmpp <iθ=δ^l₁₁¹¹···δ^l_1m^1m₁¹δ^l₂₁²¹···δ^lpm^pmpp (2.1)

if and only if the vector (ordiθ,ordθ,ord1θ, . . . ,ord_i−1θ,ord_i+1θ, . . . ,ordpθ, ki1, . . . , kim, k11, . . . , k1m₁, k21, . . . , k_i−1,mi−1, k_i+1,1, . . . , kpmp) is less than the vector (ordiθ, ordθ,ord1θ, . . . ,ordi−1θ,ordi+1θ, . . . ,ordpθ, li1, . . . , lim, l11, . . . , l1m₁, l21, . . . , li−1,m_i−1, l_i+1,1, . . . , lpm_p)with respect to the lexicographic order onN^m+p+1.

IfRis a differential ring with a basic set∆andΣ⊆R, then the intersection of all

∆-ideals ofRcontaining the setΣis, obviously, the smallest∆-ideal ofRcontaining Σ. This ideal is denoted by[Σ]. (It is clear that[Σ]is generated, as an ideal, by the set{Θξ|ξ∈Σ}.) If the setΣis finite,Σ= {ξ1, . . . , ξq}, we say that the∆-idealI=[Σ]

is finitely generated (we write this asI=[ξ1, . . . , ξq]) and callξ1, . . . , ξqdifferential or

∆-generators ofI.

A subfield K0 of the ∆-fieldK is said to be a differential (or∆-) subfield of Kif δ(K0)⊆K0for anyδ∈∆. IfK0is a∆-subfield of the∆-fieldKandΣ⊆K, then the intersection of all∆-subfields ofKcontainingK0andΣis the unique∆-subfield ofK containingK0andΣand contained in every∆-subfield ofKcontainingK0andΣ. It is denoted byK0Σ. IfK=K0Σand the setΣis finite,Σ= {η1, . . . , ηn}, thenKis said to be a finitely generated∆-extension ofK0with the set of∆-generators{η1, . . . , ηn}. In this case we writeK=K0η1, . . . , ηn. It is easy to see that the fieldK0η1, . . . , ηn coincides with the fieldK0({θηi|θ∈Θ,1≤i≤n}).

Now we can formulate the Kolchin’s theorem on differential dimension polynomial (see [7, Chapter 2, Theorem 6]). As usual,_t

k

(k∈Z, k≥1) denotes the polynomial t(t−1)···(t−k+1)/k! in one variablet,_t

0

=1, and_t

k

=0 ifk <0.

Theorem 2.1. Let K be a differential field with a basic set∆= {δ1, . . . , δm} and L a differential field extension ofK generated by a finite set η= {η1, . . . , ηn}. Then there exists a polynomialω_η|K(t)in one variabletwith rational coefficients (called a differential dimension polynomial of the extension) such that

(i) ωη|K(r )=trdeg_KK({θηj|θ∈Θ(r ),1≤j≤n})for all sufficiently large inte- gersr;

(ii) degmω_η|K ≤ m and the polynomial ω_η|K(t) can be written as ω_η|K(t) =

i=1ai

_t

+i i

, wherea0, . . . , amare some integers;

(iii) the degree dof the polynomial ω_η|K and the coefficients am andad do not depend on the choice of the system of ∆-generators η of the extensionL/K (clearly, ad ≠am if and only ifd < m, that is, am=0). In other words, d, am, andad are differential birational invariants of the extension. Moreover, the coefficientam is equal to the differential transcendence degree ofLover K, that is, to the maximal number of elementsξ1, . . . , ξk∈Lsuch that the set {θξi|θ∈Θ,1≤i≤k}is algebraically independent overK.

IfY= {y1, . . . , yn}is a finite set of symbols, then one can consider the countable set of symbolsΘY= {θyj|θ∈Θ,1≤j≤n}(calledterms) and the polynomial ringR= K[{θyj|θ∈Θ,1≤j≤n}]in the set of indeterminatesΘY over the differential field K. This polynomial ring is naturally viewed as a∆-ring whereδ(θyj)=(δθ)yj(δ∈∆, θ∈Θ, 1≤j≤n) and the elements of∆act on the coefficients of the polynomials ofR as they act in the fieldK. The ringRis called thering of differential(or∆-)polynomials in the set of differential (∆-)indeterminatesy1, . . . , ynover the∆-fieldK. This ring is denoted byK{y1, . . . , yn}and its elements are called differential (or∆-) polynomials.

The set of all termsΘY will be considered together withp orderings that corre- spond to the orderings of the semigroupΘand that are denoted by the same symbols

<1, . . . , <p. These orderings ofΘY are defined as follows:θyj<iθyk (θ, θ∈Θ,1≤ j, k≤n,1≤i≤p) if and only ifθ <iθorθ=θandj < k.

By theith order of a termu=θyjwe mean the number ordiu=ordiθ. The number ordu=ordθis called the order of the termu.

We say that a termu=θyiis divisible by a termv=θyj(oruis amultipleofv) and writev|u, ifi=jandθ|θ. For any termsu1=θ1yj, . . . , uq=θqyjcontaining the same∆-indeterminateyj(1≤j≤n), the term lcm(θ1, . . . , θq)yjis called the least common multiple ofu1, . . . , uq, it is denoted by lcm(u1, . . . , uq).

If A∈K{y1, . . . , yn},A∉K, and 1≤i≤p, then the highest with respect to the ordering<iterm that appears inAis called thei-leader of the∆-polynomialA. It is denoted byu⁽ⁱ⁾_A . IfAis written as a polynomial in one variableu⁽¹⁾_A ,A=Id(u⁽¹⁾_A )^d+ I_d−1(u⁽¹⁾_A )^d−1+ ··· +I0 (the ∆-polynomials Id, I_d−1, . . . , I0 do not contain u⁽¹⁾_A ), then Id is called theleading coefficient of the∆-polynomialAand the partial derivative

∂A/∂u⁽¹⁾_A =dId(u⁽¹⁾_A )^d−1+(d−1)I_d−1(u⁽¹⁾_A )^d−2+ ··· +I1is called theseparant ofA.

The leading coefficient and the separant of a∆-polynomialAare denoted byIAand SA, respectively.

Definition2.2. LetAandBbe two∆-polynomials fromK{y1, . . . , yn}. We say that Ahas a lower rank thanBand writer k A < r k Bif eitherA∈K,B∉K, or the vector u⁽¹⁾_A ,deg_u(1)

A A,ord2u⁽²⁾_A , . . . ,ordpu^(p)_A

is less than the vector

u⁽¹⁾_B ,deg_u(1)

B B,ord2u⁽²⁾_B , . . . ,ordpu^(p)_B

with respect to the lexicographic order (whereu⁽¹⁾_A and u⁽¹⁾_B are com- pared with respect to<1and all other coordinates of the vectors are compared with respect to the natural order onN). If the two vectors are equal (orA∈KandB∈K) we say that the∆-polynomialsAandBare of the same rank and writer k A=r k B.

LetKbe a∆-field andG=Kη1, . . . , ηna finitely generated∆-extension ofKwith a set of generators η= {η1, . . . , ηn}. Then there exists a natural ∆-homomorphism Φη from the ring of∆-polynomials K{y1, . . . , yn}toG such thatΦη(a)=afor any

a∈KandΦη(yj)=ηjforj=1, . . . , n. IfA∈K{y1, . . . , yn}, then the elementΦη(A) is called thevalueof the∆-polynomialAatη, it is denoted byA(η). Obviously, the kernelP of the mappingΦη is a prime∆-ideal of the ringK{y1, . . . , yn}. This ideal is called thedefiningideal ofηoverKor the defining ideal of the∆-field extension G=Kη1, . . . , ηn. It is easy to see that if the quotient fieldQof the factor ring ¯R= K{y1, . . . , yn}/P is considered as a∆-field (whereδ(u/v)=(vδ(u)−uδ(v))/v²for anyu, v ∈R), then¯ Qis naturally∆-isomorphic to the field G. (The appropriate∆- isomorphism is identical onKand maps the images of the∆-indeterminatesy1, . . . , yn

in the factor ring ¯Ronto the elementsη1, . . . , ηn, respectively.)

3. Numerical polynomials. A polynomialf (t1, . . . , tp)inpvariablest1, . . . , tp(p∈N, p≥1) with rational coefficients is callednumerical iff (t1, . . . , tp)∈Zfor all suffi- ciently large(t1, . . . , tp)∈Z^p, that is, there exists1, . . . , sp∈Zsuch thatf (r1, . . . , rp)∈Z as soon as(r1, . . . , rp)∈Z^pandri≥sifor alli=1, . . . , p.

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 formp

i=1

_t

i m_i

wherem1, . . . , mp∈N(p∈N, p≥1).

If f (t1, . . . , tp) is a numerical polynomial, then degf and deg_tif (1≤i≤p) will denote the total degree off and the degree of f relative to the variableti, respec- tively. The following theorem proved in [8] gives the “canonical” representation of a numerical polynomial in several variables.

Theorem3.1. Letf (t1, . . . , tp)be a numerical polynomial inpvariablest1, . . . , tp, and letdegt_if=mi(m1, . . . , mp∈N). Then the polynomialf (t1, . . . , tp)can be repre- sented in the form

f

t1, . . . , tp

=

m1

i₁=0

···

m_p

i_p=0

ai₁···ip

t1+i1

i1

··· tp+ip

ip

(3.1) with integer coefficientsai₁···ip(0≤ik≤mkfork=1, . . . , p) that are uniquely defined by the numerical polynomial.

In the rest of the section we deal with subsets ofN^mwhere the positive integerm is represented as a sum ofpnonnegative integersm1, . . . , mp(p∈N, p≥1). In other words, we fix a partition(m1, . . . , mp)of the numberm.

IfᏭ⊆N^m andr1, . . . , rp∈N, thenᏭ(r1, . . . , rp)will denote the set{(a1, . . . , am)∈ Ꮽ|a1+···+am1≤r1, am1+1+···+am1+m2≤r2, . . . , am1+···+mp−1+1+···+am≤rp}.

Furthermore,V_Ꮽwill denote the set of allm-tuplesv=(v1, . . . , vm)∈N^mthat are not greater than or equal to anym-tuple fromᏭwith respect to the product order onN^m. (Recall that the product order on the setN^k(k∈N, k≥1) is a partial order≤P such thatc=(c1, . . . , ck)≤Pc=(c₁, . . . , c_k)if and only ifci≤c_ifor alli=1, . . . , k. Ifc≤Pc andc≠c, we writec <Pc). Clearly, an elementv=(v1, . . . , vm)∈N^mbelongs toV_Ꮽ if and only if for any(a1, . . . , am)∈Ꮽ, there existsi∈N, 1≤i≤m, such thatai> vi. The following two theorems proved in [8] generalize the well-known Kolchin’s result on the numerical polynomials associated with subsets ofN(see [7, Chapter 0, Lemma 17]) and give the explicit formula for the numerical polynomials inpvariables associated with a finite subset ofN^m.

Theorem3.2. LetᏭbe a subset ofN^mwherem=m1+···+mpfor some nonnega- tive integersm1, . . . , mp(p≥1). Then there exists a numerical polynomialω_Ꮽ(t1, . . . , tp) with the following properties:

(i) ω_Ꮽ(r1, . . . , rp)=CardV_Ꮽ(r1, . . . , rp)for all sufficiently large(r1, . . . , rp)∈N^p(as usual,CardMdenotes the number of elements of a finite setM).

(ii) degω_Ꮽ≤manddeg_tiω_Ꮽ≤mifori=1, . . . , p.

(iii) degω_Ꮽ=mif and only ifᏭ= ∅. In this case,ω_Ꮽ(t1, . . . , tp)=p i=1

_t

i+m_i m_i

. (iv) ω_Ꮽis a zero polynomial if and only if(0, . . . ,0)∈A.

The polynomialω_Ꮽ(t1, . . . , tp)whose existence is stated byTheorem 3.2is called thedimension polynomial of the setᏭ⊆N^massociated with the partition(m1, . . . , mp) ofm. Ifp=1, the polynomialω_Ꮽis called theKolchin polynomialof the setᏭ.

Theorem3.3. LetᏭ= {a1, . . . , an}be a finite subset ofN^mand(m1, . . . , mp) (p≥1) a partition ofm. Letai=(ai1, . . . , aim) (1≤i≤n)and for anyl∈N,0≤l≤n, let Γ(l, n)denote the set of alll-element subsets of the setNn= {1, . . . , n}. Furthermore, for anyσ∈Γ(l, p), leta¯σ j=max{aij|i∈σ}(1≤j≤m) andbσ j=

h∈σ ja¯σ h. Then

ω_Ꮽ

t1, . . . , tp

= n l=0

(−1)^l

σ∈Γ(l,p)

n j=1

tj+mj−bσ j

mj

. (3.2)

It is clear that ifᏭ⊆N^m and Ꮽis the set of all minimal elements of the setᏭ with respect to the product order onN^m, then the setᏭis finite andω_Ꮽ(t1, . . . , tp)= ω_Ꮽ(t1, . . . , tp). Thus,Theorem 3.3gives an algorithm that allows to find a numeri- cal polynomial associated with any subset ofN^m(and with a given partition ofm):

one should first find the set of all minimal points of the subset and then apply Theorem 3.3.

4. Reduction in the ring of differential polynomials. In what follows we keep the notation and conventions ofSection 2. In particular,K{y1, . . . , yn}denotes the ring of∆-polynomials over a differential fieldKwhose basic set∆is a union ofpdisjoint sets:∆=∆1

···

∆p, where∆i= {δi1, . . . , δim_i}(1≤i≤p).

Definition4.1. LetA, B∈K{y1, . . . , yn}andA∉K. The∆-polynomialBis said to be reduced with respect toAif the following two conditions hold:

(i) Bdoes not contain any termθu⁽¹⁾_A (θ∈Θ, θ≠1)such that ordi(θu⁽ⁱ⁾_A )≤ordiu⁽ⁱ⁾_B fori=2, . . . , p.

(ii) IfB containsu⁽¹⁾_A , then either ordju^(j)_B <ordju^(j)_A for some j (2≤j ≤p) or ordiu⁽ⁱ⁾_A ≤ordiu⁽ⁱ⁾_B for alli=2, . . . , pand deg_u(1)

A B <deg_u(1) A A.

A ∆-polynomial B is said to be reduced with respect to a set of∆-polynomials Σ⊆K{y1, . . . , yn}ifBis reduced with respect to every element ofΣ.

Definition4.2. A set of ∆-polynomialsΣ⊆K{y1, . . . , yn}is called autoreduced ifΣ

K= ∅and every element ofΣis 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].

Lemma4.3. LetNn= {1, . . . , n}and letAbe an infinite subset ofN^m×Nn(m, n∈N, n≥1). Then there exists an infinite sequence of elements ofA, strictly increasing relative to the product order, in which every element has the same projection onNn.

This result implies the following statement that will be used below.

Lemma4.4. LetS be an infinite set of terms in the ringK{y1, . . . , yn}. Then there exists an indexj (1≤j≤n)and an infinite sequence of termsθ1yj, θ2yj, . . . , θkyj, . . .∈S such thatθk|θk+1for allk=1,2, . . . .

Theorem4.5. Every autoreduced set of∆-polynomials is finite.

Proof. Suppose thatΣis an infinite autoreduced subset of K{y1, . . . , yn}. Then Σcontains an infinite set Σ such that all∆-polynomials fromΣ have different 1- leaders. Indeed, if it is not so, then there exists an infinite set Σ1 ⊆Σ such that all∆-polynomials fromΣ1 have the same 1-leaderu. ByLemma 4.3, the infinite set {(ord2u⁽²⁾_A , . . . ,ordpu^(p)_A ) | A ∈ Σ1} contains a nondecreasing infinite sequence (ord2u⁽²⁾_A1, . . . ,ordpu^(p)_A1)≤P (ord2u⁽²⁾_A2, . . . ,ordpu^(p)_A2)≤P ··· (A1, A2, . . .∈Σ1 and ≤P

denotes the product order onN^p−1). Since the sequence{deguAi|i=1,2, . . .}cannot strictly decrease, there exists two indicesiandjsuch thati < jand deg_uAi≤deg_uAj. We obtain thatAjis reduced with respect toAithat contradicts the fact thatΣis an autoreduced set.

Thus, we can assume that elements ofΣ have distinct 1-leaders. By Lemma 4.4, there exists an infinite sequenceB1, B2, . . .of elements ofΣsuch thatu⁽¹⁾_B

i |u⁽¹⁾_Bi+1 for i=1,2, . . . .Letkij =ordju⁽¹⁾_Bi andlij=ordju^(j)_Bi (2≤j≤p). Obviously,lij≥kij (i= 1,2, . . .;j=2, . . . , p), so that{(li2−ki2, . . . , lip−kip)|i=1,2, . . .} ⊆N^p−1. ByLemma 4.3, there exists an infinite sequence of indicesi1< i2<···such that(li₁2−ki₁2, . . . , li₁p− ki1p)≤P(li22−ki22, . . . , li2p−ki2p)≤P. . . .Then for anyj=2, . . . , p, we have

ordj



u⁽¹⁾_Bi

2

u⁽¹⁾_Bi

1

u^(j)_B

i1



=ki₂j−ki₁j+li₁j≤ki₂j+li₂j−ki₂j=li₂j=ordju^(j)_B

i2, (4.1)

so thatBi₂ contains a termθu⁽¹⁾_B

i1=u⁽¹⁾_B

i2 such thatθ≠1 and ordj(θu^(j)_B

i1)≤ordju^(j)_B

i2

forj=2, . . . , p. Thus, the∆-polynomialBi2is reduced with respect toBi1that contra- dicts the fact thatΣis an autoreduced set. This completes the proof of the theorem.

Below, the elements of an autoreduced set will be always arranged in order of in- creasing rank. (Therefore, if we consider an autoreduced set of∆-polynomialsΣ= {A1, . . . , Ar}, thenr k A1<···< r k Ar.)

The following two theorems can be proven precisely in the same way as their clas- sical analogs (see [7, Chapter 1, Corollary to Lemma 6 and Proposition 3, page 78]).

Theorem4.6. LetΣ= {A1, . . . , Ar}be an autoreduced set in the ringK{y1, . . . , yn} and letBbe a∆-polynomial. Then there exist a∆-polynomialB0and nonnegative in- tegers pi, qi (1≤i≤r) such thatB0 is reduced with respect to Σ, r k B0≤r k B, and r

i=1I_A^pi_iS_A^qi_iB≡B0(mod[Σ]).

Definition4.7. LetΣ= {A1, . . . , Ar}andΣ= {B1, . . . , Bs}be two autoreduced sets in the ring of differential polynomialsK{y1, . . . , yn}. An autoreduced setΣis said to have lower rank thanΣif one of the following two cases holds:

(1) there existsk∈Nsuch thatk≤min{r , s},r k Ai=r k Bifori=1, . . . , k−1 and r k Ak< r k Bk;

(2) r > sandr k Ai=r k Bifori=1, . . . , s.

Ifr=sandr k Ai=r k Bifori=1, . . . , r, thenΣis said to have the same rank asΣ. Theorem4.8. In every nonempty family of autoreduced sets of differential polyno- mials there exists an autoreduced set of lowest rank.

LetJbe an ideal of the ringK{y1, . . . , yn}. Since the family of all autoreduced subsets ofJis not empty (e.g., it contains the empty set),Theorem 4.8shows that the idealJ contains an autoreduced subset of lowest rank.

Definition4.9. LetJbe an ideal of the ring of differential polynomialsK{y1, . . . , yn}. Then an autoreduced subset ofJof lowest rank is called a characteristic set of the idealJ.

Theorem4.10. LetΣ= {A1, . . . , Ad}be a characteristic set of an idealJof the ring of∆-polynomialsR=K{y1, . . . , yn}. Then an elementB∈Ris reduced with respect to the setΣif and only ifB=0.

Proof. Suppose thatB≠0. ThenB and elements ofΣwhose rank is lower than the rank ofB form an autoreduced setΣ. It is easy to see thatΣhas a lower rank thanΣthat contradicts the fact thatΣis a characteristic set of the idealJ.

Theorem4.11. LetJbe a cyclic differential ideal of the ring of∆-polynomialsR= K{y1, . . . , yn}generated by a linear∆-polynomialf. Then{f}is a characteristic set of the∆-idealJ=[f ].

Proof. First of all, we show that no nonzero element ofJis reduced with respect tof. Let 0≠h∈J, and let k be the smallest positive integer such thath can be written as

h=g1θ1f+···+gkθkf , (4.2) for some pairwise distinct elementsθ1, . . . , θk∈Θ and someg1, . . . , gk∈R. In what follows, we suppose thatk >1 (clearly, an element of the formgθf (g∈R, θ∈Θ) is not reduced with respect tof) andθ1<1···<1θk. Furthermore, it is obvious that θkfgjforj=1, . . . , k−1 (otherwise,his a linear combination ofk−1 elements from Θf).

Sinceθkf is linear with respect toθku⁽¹⁾_f , one can write each∆-polynomialgj(1≤ j≤k−1) asgj=g_j+g_j(θkf ), whereg_j, g_j ∈Randg_jdoes not containθku⁽¹⁾_f . Then h=g₁θ1f+···+g_k−1θ_k−1f+gθkffor someg∈R.

Sincegθkf containsθku⁽¹⁾_f and none ofg_jθjf (1≤j≤k−1) contains this term, the∆-polynomialhcontainsθku⁽¹⁾_f andr k f≤r k h. Similarly, ifθj_i is the maximal element of the set{θ1, . . . , θk}relative to the order<ionΘ(2≤i≤p), thenhcontains θj_iu⁽ⁱ⁾_f , so that ordi(θku⁽ⁱ⁾_f )≤ordi(θj_iu⁽ⁱ⁾_f )≤ordiu⁽ⁱ⁾_h . It follows thathis reduced with

respect tofandfis an element of the lowest rank inJ. Therefore, ifΣ= {h1, . . . , hl}is a characteristic set ofJ, thenr k f=r k h1andl=1, whence{f}is also a characteristic set of the idealJ.

5. 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 (seeTheorem 2.1).

Theorem5.1. LetKbe a differential field whose basic set of derivation operators∆ is a union ofpdisjoint finite sets (p≥1):∆=∆1

···

∆p, where∆i= {δi1, . . . , δim_i} (m1, . . . , mpare some positive integers). Furthermore, letG=Kη1, . . . , ηnbe a∆-field extension ofKgenerated by a finite setη= {η1, . . . , ηn}. Then there exists a polynomial Φη(t1, . . . , tp)inpvariablest1, . . . , tpwith rational coefficients such that

(i) Φη(r1, . . . , rp)=trdeg_KK(n

j=1Θ(r1, . . . , rp)ηj)for all sufficiently large(r1, . . . , rp)∈Z^p;

(ii) deg_t

iΦη≤mi(i=1, . . . , p) and the polynomialΦη(t1, . . . , tp)can be written as Φη

t1, . . . , tp

=

m1

i₁=0

···

m_p

ip=0

ai₁···ip

t1+i1

i1

··· tp+ip

ip

, (5.1)

whereai₁···ip∈Zfor alli1, . . . , ip.

Proof. LetP be the defining ∆-ideal of the extension G/Kand Σ= {A1, . . . , Ad} a characteristic set ofP. Furthermore, for anyr1, . . . , rp∈N, letUr₁···r_p= {u∈ΘY | ordiu≤rifori=1, . . . , pand eitheruis not a multiple of anyu⁽¹⁾_Ai, or for everyθ∈Θ, A∈Σsuch thatu=θu⁽¹⁾_A , there existsi∈ {2, . . . , p}such that ordi(θu⁽ⁱ⁾_A ) > ri}. (If p=1, we setUr₁= {u∈ΘY|ord1u≤r1anduis not a multiple of anyu⁽¹⁾_A

i}.) We are going to show that the set ¯Ur₁···rp= {u(η)|u∈Ur₁···rp}is a transcendence basis of the fieldK(n

j=1Θ(r1, . . . , rp)ηj)overK.

First of all, we show that the set ¯Ur₁···rpis algebraically independent overK. Letgbe a polynomial inkvariables (k∈N,k≥1) such thatg(u1(η), . . . , uk(η))=0 for some elementsu1, . . . , uk∈Ur₁···r_p. Then the∆-polynomial ¯g=g(u1, . . . , uk)is reduced with respect toΣand ¯g∈P. ApplyingTheorem 4.10we obtain that ¯g=0. Thus, the set U¯r₁···rpis algebraically independent overK.

Now, we show that every elementθηj(1≤j≤n, θ∈Θ(r1, . . . , rp)) is algebraic over the fieldK(U¯r₁,...,r_p). Letθηj∉U¯r₁,...,r_p (ifθηj∈U¯r₁,...,r_p, the statement is obvious).

Thenθyj∉Ur1,...,rp whenceθyj is equal to some term of the formθu⁽¹⁾_Ai (θ∈Θ, 1≤i≤d) such that ordk(θu^(k)_A

i)≤rkfork=2, . . . , p. We representAias a polynomial inu⁽¹⁾_A

i :Ai=I0(u⁽¹⁾_A

i)^e+I1(u⁽¹⁾_A

i)^e−1+ ··· +Ie, whereI0,I1, . . . , Ie do not containu⁽¹⁾_A

i

(therefore, all terms in these∆-polynomials are lower than u⁽¹⁾_A

i with respect to the order<1). SinceAi∈P,

Ai(η)=I0(η) u⁽¹⁾_A

i(η)e

+I1(η) u⁽¹⁾_A

i(η)e−1

+···+Ie(η)=0. (5.2) It is clear that I0 and SA_i = ∂Ai/∂u⁽¹⁾_Ai are reduced with respect to the set Σ. By

Theorem 4.10,I0∉P and SA_i ∉P whenceI0(η)≠0 andSA_i(η)≠0. If we apply the operatorθto both sides of (5.2), the resulted equation will show that the element θu⁽¹⁾_A

i(η)=θηj is algebraic over the fieldK({θη¯ l|ordiθ¯≤ri(1≤i≤p, 1≤l≤n) and ¯θyl<1θu⁽¹⁾_A

i =θyj}). Now, the induction on the setΘY ordered by the relation

<1completes the proof of the fact that ¯Ur₁···r_p(η)is a transcendence basis of the field K(n

j=1Θ(r1, . . . , rp)ηj)overK.

LetUr⁽¹⁾₁···r_p = {u∈ΘY |ordiu≤ri fori=1, . . . , p and u≠θu⁽¹⁾_Aj for anyθ∈Θ;

j=1, . . . , d}andUr⁽²⁾₁···rp= {u∈ΘY|ordiu≤rifori=1, . . . , pand there exists at least one pairi,j (1≤i≤p,1≤j≤d)such thatu=θu⁽¹⁾_Aj and ordi(θu⁽ⁱ⁾_Aj) > ri}. (Ifp=1, then we setUr⁽²⁾₁···rp= ∅.) Clearly,Ur₁···r_p=Ur⁽¹⁾₁···rp

Ur⁽²⁾₁···rpandUr⁽¹⁾₁···rp

Ur⁽²⁾₁···rp= ∅. By Theorem 3.2, there exists a numerical polynomial ω(t1, . . . , tp)in p variables t1, . . . , tpsuch thatω(r1, . . . , rp)=CardUr⁽¹⁾₁···rpfor all sufficiently large(r1, . . . , rp)∈Z^p and deg_tiω≤mi(i=1, . . . , p). Thus, in order to complete the proof of the theorem, we need to show that there exists a numerical polynomialφ(t1, . . . , tp)inpvariables t1, . . . , tpsuch thatφ(r1, . . . , rp)=CardUr⁽²⁾1···rpfor all sufficiently large(r1, . . . , rp)∈Z^p and deg_tiφ≤mi(i=1, . . . , p).

Let ordiu⁽¹⁾_Aj =aij and ordiu⁽ⁱ⁾_Aj =bij fori=1, . . . , p;j=1, . . . , d(clearly,aij≤bij

anda1j=b1jfori=1, . . . , p;j=1, . . . , d). Furthermore, for anyq=1, . . . , pand for any integersk1, . . . , kqsuch that 1≤k1<···< kq≤p, letVj;k₁,...,kq(r1, . . . , rp)= {θu⁽¹⁾_A

j | ordiθ≤ri−aij fori=1, . . . , pand ordkθ > rk−bkj if and only ifkis equal to one of the numbersk1, . . . , kq}. Then CardVj;k₁,...,k_q(r1, . . . , rp)=φj;k₁,...,k_q(r1, . . . , rp), where φj;k₁,...,k_q(t1, . . . , tp)is a numerical polynomial inp variablest1, . . . , tpdefined by the formula

φj;k₁,...,k_q

t1, . . . , tp

= t1+m1−a1j

m1

··· tk₁−1+mk₁−1−ak₁−1,j

mk₁−1

× tk₁+mk₁−ak₁,j

mk₁

− tk₁+mk₁−bk₁,j

mk₁

× tk₁+1+mk₁+1−ak₁+1,j

mk1+1

··· tk_q−1+mk_q−1−ak_q−1,j

mkq−1

× tk_q+mk_q−ak_q,j

mkq

− tk_q+mk_q−bk_q,j

mkq

×··· tp+mp−apj

mp

.

(5.3)

(ByTheorem 3.2(iii), Card{θ∈Θ|ord1θ≤s1, . . . ,ordpθ≤sp} =p i=1

_s

i+mi m_i

for any s1, . . . , sp∈N.) Clearly, deg_tiφj;k₁,...,kq≤mifori=1, . . . , p.

Now, for anyj=1, . . . , d, letVj(r1, . . . , rp)= {θu⁽¹⁾_Aj |ordiθ≤ri−aij fori=1, . . . , p and there existsk∈N, 1≤k≤p, such that ordkθ > rk−bkj}. Applying the com- binatorial principle of inclusion and exclusion we obtain that CardVj(r1, . . . , rp)=