Dimension of Finitely Generated Differential and Difference Field - - PDF document

Dimension of Finitely Generated Differential and Difference Field Extensions

Alexander Levin

The Catholic University of America Washington, D. C. 20064 E-mail: levin@cua.edu

Fourth International Workshop on Differential Algebra and Related Topics October 27 - 30, 2010 Beijing, China

1

• Let K denote either a differential field with

basic set of derivations ∆ = {δ1, . . . , δm} or an inversive difference field with basic set of automorphisms σ = {α1, . . . , αm}. We assume that Char K = 0. In the first case, we say that K is a ∆-field, in the second case we call K a σ∗-field setting σ∗ = {α1, . . . , αm, α−1

1 , . . . , α−1 m }.

2

• If K is a ∆-field, then Θ will denote the

free commutative semigroup generated by ∆; we define the order of θ = δk1

1 . . . δkm m

∈ Θ (ki ∈ N) as ord θ =

m

• i=1

ki and set Θ(r) = {θ ∈ Θ | ord θ ≤ r} for any r ∈ N.

• If K is a σ∗-field, then Γ will denote the free

commutative group generated by σ; the order

• f an element γ = αl1

1 . . . αlm m ∈ Γ (li ∈ Z) is

defined as ord γ =

m

• i=1

|li| and for any r ∈ N, we set Γ(r) = {γ ∈ Γ | ord γ ≤ r}.

3

• Let L = Kη1, . . . , ηn be a ∆- or σ∗- field

extension of K generated by a finite set η = {η1, . . . , ηn}. As a field, L = K({θηj|θ ∈ Θ, 1 ≤ j ≤ n}) in the differential case and L = K({γηj|γ ∈ Γ, 1 ≤ j ≤ n}) in the difference case. The following statement is the classical Kolchin theorem on differential dimension polynomial. ([Kolchin, E. R. The notion of dimension in the theory of algebraic differential equations.

• Bull. AMS., 70 (1964), 570 - 573.]

4

Theorem 1. With the above notation, there exists a polynomial ωη|K(t) ∈ Q[t] such that (i) ωη|K(r) = tr.degKK({θηj|θ ∈ Θ(r), 1 ≤ j ≤ n}) for all sufficiently large r ∈ Z; (ii) deg ωη|K ≤ m and ωη|K(t) can be written as ωη|K(t) =

m

• i=0

ai t + i i

• where ai ∈ Z.

(iii) d = deg ωη|K, am and ad do not depend

• n the set of differential generators η of L/K

(ad = am iff d < m). Moreover, am is equal to the differential (∆-) transcendence degree of L

• ver K (denoted by ∆-tr.degKL), that is, to

the maximal number of elements ξ1, . . . , ξk ∈ L such that the family {θξi|θ ∈ Θ, 1 ≤ i ≤ k} is algebraically independent over K.

5

• The polynomial ωη|K(t) is called the differ-

ential dimension polynomial of the ∆-field extension L/K associated with the set of ∆- generators η = {η1, . . . , ηn}.

• The numbers d = deg ωη|K and ad are

called the differential (or ∆-) type and typi- cal differential (or ∆-) transcendence degree

• f the extension L/K; they are denoted by

∆-typeKL and ∆-t.tr.degKL, respectively.

• Methods and examples of computation of

differential dimension polynomials can be found in [Kondrateva, M. V.; Levin, A. B.; Mikhalev,

• A. V.; Pankratev, E. V. Differential and Differ-

ence Dimension Polynomials. Kluwer Acad. Publ., 1999.]

6

Theorem 1 allows one to assign numerical polynomials to certain systems of algebraic dif- ferential equations as follows. Let R = K{y1, . . . , yn} be the algebra of differential (∆-) polynomials over the ∆-field

• K. (Recall that R is the polynomial ring

K[{θyj|θ ∈ Θ, 1 ≤ j ≤ n}] in a denumer- able set of indeterminates θyj treated as a dif- ferential ring extension of K where δ(θyj) = (δθ)yj.) By a system of algebraic differential equa- tions over K we mean a system of the form fi(y1, . . . , yn) = 0 (i ∈ I) where {fi}i∈I ⊆ R; by a solution we mean an n-tuple with coordinates in some differential field extension of K that annuls all fi.

7

Let P be the differential ideal generated by {fi|i ∈ I} in R. If it is prime, then Q(R/P) = Kη1, . . . , ηn where ηj is the image of yj in R/P. By Theorem 1, we obtain a numerical polynomial ωη|K(t) called the differential di- mension polynomial of the system. This polynomial can be viewed as the alge- braic version of the A. Einstein’s strength of a system of PDEs governing a physical field defined as follows (A. Einstein): ”... the system of equations is to be chosen so that the field quantities are determined as strongly as possible. In order to apply this principle, we propose a method which gives a measure of strength of an equation system. We expand the field variables, in the neighborhood

• f a point P, into a Taylor series (which pre-

supposes the analytic character of the field);

8

the coefficients of these series, which are the derivatives of the field variables at P, fall into sets according to the degree of differentiation. In every such degree there appear, for the first time, a set of coefficients which would be free for arbitrary choice if it were not that the field must satisfy a system of differential equations. Through this system of differential equations (and its derivatives with respect to the coordi- nates) the number of coefficients is restricted, so that in each degree a smaller number of co- efficients is left free for arbitrary choice. The set of numbers of ”free” coefficients for all de- grees of differentiation is then a measure of the ”weakness” of the system of equations, and through this, also of its ”strength”. ” The following result provides an essential gen- eralization of Kolchin’s theorem.

9

Theorem 2 (L., 2009). Let K be a ∆-field (Char K = 0, ∆ = {δ1, . . . , δm}) and let L = Kη1, . . . , ηn be a ∆-field extension of

• K. Let F be an intermediate differential field
• f the extension L/K and for any r ∈ N, let

Fr = F K({θηj|θ ∈ Θ(r), 1 ≤ j ≤ n}). Then there exists a polynomial ωK,F,η(t) ∈ Q[t] such that (i) ωK,F,η(r) = tr.degKFr for all sufficiently large r ∈ Z; (ii) deg ωK,F,η ≤ m and ωK,F,η(t) can be written as ωK,F,η(t) =

m

• i=0

bi t + i i

• where bi ∈ Z .

(iii) d = deg ωK,F,η(t), bm and bd do not depend on the set of ∆-generators η of L/K. Furthermore, bm = ∆-tr.degKF.

10

If F = L, then Theorem 2 gives the Kolchin

• theorem. Furthermore, Theorem 2 shows that

Einstein’s strength of a system of algebraic differential equations, whose solution should be invariant with respect to the action of any group G commuting with basic operators δi, is expressed by a polynomial function. (We mean that δiG = Gδi for i = 1, . . . , m and g(a) = a for any g ∈ G, a ∈ K.) Indeed, in this case the fixed field F of the group G is an intermediate ∆-field of the corre- sponding ∆-field extension L/K, so the poly- nomial ωK,F,η(t) (where η is a system of dif- ferential generators of L/K) expresses A. Ein- stein’s strength of the system in this sense.

11

Note that if an intermediate field E of the extension L/K is not differential, there might be no polynomial whose values at sufficiently large r ∈ Z are equal to tr.degK(E K({θηj|θ ∈ Θ(r), 1 ≤ j ≤ n})) Indeed, let K be an ordinary differential field with one basic derivation δ, let L = Ky be the differential field of fractions of one differ- ential indeterminate y over K, and let E = K(δ2y, δ4y, . . . , δ2ky, . . . ). Then tr.degK(E

• K({θy|θ ∈ Θ(r)})) = [log2 r].

The following result gives a dimension poly- nomial for a finitely generated inversive differ- ence field extension.

12

Theorem 3 (L., 1980). Let K be an inver- sive difference (σ∗-) field (σ = {α1, . . . , αm}, Char K = 0) and let L = Kη1, . . . , ηn. Then there exists a polynomial φη|K(t) ∈ Q[t] with the following properties. (i) φη|K(r) = tr.degKK({γηj|γ ∈ Γ(r), 1 ≤ j ≤ n}) for all sufficiently large r ∈ N. (ii) deg φη|K(t) ≤ m and the polynomial φη|K(t) can be written as φη|K(t) =

m

• i=0

ai2i t + i i

• where ai ∈ Z.

(iii) The integers am, d = deg φη|K and ad do not depend on η. Also, am = σ-tr.degKL, the difference transcendence degree of L/K (this is the maximal number of elements ξ1, . . . , ξk ∈ L such that the set {γξi|γ ∈ Γ, 1 ≤ i ≤ k} is algebraically independent over K).

13

The polynomial φη|K(t) is called the σ∗- dimension polynomial of the σ∗-field exten- sion L/K associated with the system of σ∗- generators η. The numbers d = deg φη|K and the coefficient ad are called the inversive dif- ference (or σ∗-) type and typical inversive dif- ference (or typical σ∗-) transcendence degree

• f L over K. They are denoted by σ∗-typeKL

and σ∗-t.trdegKL, respectively. Note that if the generators η1, . . . , ηn are σ- algebraically independent over K and φη|K(t) is the corresponding σ∗-dimension polynomial

• f L/K, then

φη|K(t) = n

m

• k=0

(−1)m−k2k m k t + k k

• .

14

Let R = K{y1, . . . , yn}∗ be the ring of in- versive difference (σ∗-) polynomials over K, that is, a polynomial ring K[{γyj|γ ∈ Γ, 1 ≤ j ≤ n}] in the denumerable set of indetermi- nates γyj treated as a σ∗-ring extension of K where γ′(γyj) = (γ′γ)yj. By a system of algebraic difference equations

• ver K we mean a system of the form

fi(y1, . . . , yn) = 0 (i ∈ I) where {fi}i∈I ⊆ R; a solution of such a sys- tem is an n-tuple with coordinates in some σ∗-overfield of K that annuls all fi. If the the σ∗-polynomials {fi} generate a prime σ∗-ideal P in R, then Q(R/P) = Kη1, . . . , ηn where ηj is the image of yj in R/P. By Theorem 3, we obtain a numerical polynomial φη|K(t) called the σ∗-dimension polynomial of the system.

15

As in the differential case, such a polynomial expresses the strength of a system of equations in finite differences in the following sense: Consider a system of such equations with re- spect to n unknown grid functions y1, . . . , yn

• f m real variables defined at the nodes of a

grid with with cells of size h1 × · · · × hm. Let αi : y(x1, . . . , xm) → y(x1, . . . , xi+hi, . . . , xm) (1 ≤ i ≤ m). Then the number of independent values of the solutions of the system at the nodes of

• rder at most r with respect to some fixed

node O is expressed by the corresponding σ∗- dimension polynomial, so this polynomial can be treated as the strength of the system of equations in finite differences in the sense of

• A. Einstein.

16

The following is a ′′σ∗-version′′ of Theorem 2. Theorem 4. Let K be a σ∗-field (Char K = 0, σ = {α1, . . . , αm}) and L = Kη1, . . . , ηn a σ∗-field extension of K. Let F be an in- termediate σ∗-field of the extension L/K and for any r ∈ N, let Fr = F K({γηj|γ ∈ Γ(r), 1 ≤ j ≤ n}). Then there exists a poly- nomial φK,F,η(t) ∈ Q[t] such that (i) φK,F,η(r) = tr.degKFr for all sufficiently large r ∈ Z; (ii) deg φK,F,η ≤ m and φK,F,η(t) can be written as φK,F,η(t) =

m

• i=0

2ibi t + i i

• where bi ∈ Z .

(iii) d = deg φK,F,η(t), bm and bd do not depend on the set of σ∗-generators η of L/K. Furthermore, bm = σ-tr.degKF.

17

• This theorem, in particular, gives a polyno-

mial that describes the strength of a system of algebraic difference equations whose solution should be invariant with respect to the action

• f a group G commuting with basic translation

αi.

• The following results show that some invari-

ants of differential and difference dimension polynomials can be viewed as Krull-type di- mensional characteristics of the corresponding field extensions. Let K be either a differential (∆-) field with basic set ∆ = {δ1, . . . , δm} or an inversive dif- ference (σ∗-) field with basic set σ = {α1, . . . , αm}. Let L = Kη1, . . . , ηn and let U denote the set of all intermediate ∆- (respectively, σ∗-) fields of the extension L/K.

18

Let BU = {(F, E) ∈ U × U | F ⊇ E} and let Z denote the ordered set Z {∞} (where the natural order on Z is extended by the condition a < ∞ for any a ∈ Z). Lemma 1. With the above notation, there exists a unique mapping µU : BU → Z such that (i) µU(F, E) ≥ −1 for any (F, E) ∈ BU ; (ii) If d ∈ N, then µU(F, E) ≥ d if and only if tr.degEF > 0 and there exists an infinite descending chain of intermediate ∆- (respec- tively, σ∗-) fields F = F0 ⊇ F1 ⊇ · · · ⊇ Fr ⊇ · · · ⊇ E such that µU(Fi, Fi+1) ≥ d − 1 (i = 0, 1, . . . ).

19

Note that µU(F, E) = −1 iff the field exten- sion F/E is algebraic. With the notation of Lemma 1, we define the ∆- (respectively, σ∗-) transcendental type of the ∆- (respectively, σ∗-) field extension L/K as sup{µU(F, E) | (F, E) ∈ BU}. It will be denoted by ∆-tr.type(L/K) (re- spectively, by σ∗-tr.type(L/K)). Furthermore, we define the ∆- (respectively, σ∗-) transcendence dimension of the exten- sion L/K as sup{q ∈ N | ∃ F0 ⊇ F1 ⊇ · · · ⊇ Fq such that Fi ∈ U and µU(Fi−1, Fi) = ∆-(respectively, σ∗-) tr.type(L/K) for i = 1, . . . , q}. It will be denoted by ∆-tr.dim(L/K) (re- spectively, by σ∗-tr.dim(L/K)).

20

Clearly, if ∆-tr.type(L/K) < ∞, then ∆-tr.dim(L/K) > 0. The following result is formulated for differ- ential field extensions. It remains true if one replaces every entry of ∆ by σ∗. Theorem 5. Let K be a ∆-field of zero characteristic, Card ∆ = m. Let L be a finitely generated ∆-field extension of K. Then (i) ∆-tr.type(L/K) ≤ ∆-typeKL ≤ m. (ii) If ∆-tr.degKL > 0, then ∆-tr.type(L/K) = m and ∆-tr.dim(L/K) = ∆-tr.degKL. (iii) If ∆-tr.degKL = 0, then ∆-tr.type(L/K) < m.

21

In 1969 J. Johnson proved that if L/K is a finitely generated differential field extension with a basic set ∆ = {δ1, . . . , δm} and ∆-tr.degKL = 0, then there is a linear trans- formation of the basic set ∆ into a set ∆′ = {δ′

1, . . . , δ′ m} (δ′ i = m j=1 cijδj where cij are

constants of K) such that L is a finitely gen- erated {δ′

1, . . . , δ′ m−1}-field extension of K.

The problem of reduction of the basic set of automorphisms in the difference cased can be set as follows. Let (K, σ) and (K, σ1) denote inversive dif- ference fields with the same underlying field K and with the basic sets σ = {α1, . . . , αm} and σ1 = {τ1, . . . , τm}, respectively.

22

The sets σ and σ1 are said to be equivalent if there exists a matrix A = (kij)1≤i,j≤m ∈ GL(m, Z) such that αi = τki1

1

. . . τkim

m

(1 ≤ i ≤ m). In this case we write σ ∼ σ1 and say that the transformation of the set σ into the set σ1 is an admissible transformation of σ. The following example shows that one can- not expect that if L is a finitely generated σ∗- field extension of an inversive difference field K with a basic set σ = {α1, . . . , αm}, then there exists a set σ1 = {β1, . . . , βm} of pairwise commuting automorphisms of L such that σ1 is equivalent to σ and L is a finitely generated σ∗

2-field extension of K if L and K are treated

as inversive difference fields with the basic set σ2 = {β1, . . . , βm−1}.

23

Example 1. Let K be an ordinary σ∗-field with σ = {α} and let L = Kη where the σ∗-generator η is transcendental over K and satisfies the defining equation α(η)2 = η. The corresponding σ∗-dimension polynomial φη|K(t) = 1 (hence σ-tr.degKL = 0), since tr.degKK(η, α(η), α−1(η) . . . , αr(η), α−r(η)) = 1 for all r ∈ N. At the same time L is not a finitely generated field extension of K (it is easy to check that αr(η) / ∈ K(η, α(η), . . . , αr−1(η), α−(r−1)(η)) for any r ≥ 1). However, one can obtain the following result

• n inversive difference field extensions of zero

difference transcendence degree.

24

Theorem 6. Let K be an inversive differ- ence field with a basic set σ = {α1, . . . , αm}, let L be a finitely generated σ∗-field extension

• f K, and let d = σ∗-typeKL. Then there ex-

ists a set σ1 = {β1, . . . , βm} of mutually com- muting automorphisms of L and a finite family ζ = {ζ1, . . . , ζq} of elements of L such that σ1 is equivalent to σ and if σ2 = {β1, . . . , βd}, then L is an algebraic extension of the field H = Kζ1, . . . , ζqσ2. (The last field is a finitely generated σ∗

2-field extension of K when

K is treated as an inversive difference field with the basic set σ2.)

25

The following considerations lead to an es- sential generalization of Theorems 1 - 4 on differential and difference dimension polyno-

• mials. We will consider the differential case

and show that there is a multivariate numer- ical polynomial associated with any partition

• f the basic set of derivation operators. These

multivariate differential dimension polynomi- als represent the ”generalized” strength of a system of algebraic differential equations, which is defined in the same way as the Einstein’s concept of strength if one imposes separate restrictions on the orders of derivations with respect to each group of basic derivation oper- ators.

26

Let K be a differential field (Char K = 0) whose basic set ∆ is a union of p disjoint finite sets (p ≥ 1): ∆ = ∆1 · · · ∆p, where ∆i = {δi1, . . . , δimi} (i = 1, . . . , p). Thus, we fix a partition of the set ∆. For any θ = δk11

11 . . . δ k1m1 1m1 δk21 21 . . . δ kpmp pmp ∈ Θ,

we define the order of the element θ with re- spect to ∆i as follows: ordi θ = mi

j=1 kij

(i = 1, . . . , p). Furthermore, for any r1, . . . , rp ∈ N, we set Θ(r1, . . . , rp) = {θ ∈ Θ|ordi θ ≤ ri for i = 1, . . . , p}.

27

Theorem 7 (L, 2007). Let L = Kη1, . . . , ηn be a ∆-field extension generated by a set η = {η1, . . . , ηn}. Then there exists a polynomial Φη(t1, . . . , tp) in p variables t1, . . . , tp with ra- tional coefficients such that (i) Φη(r1, . . . , rp) = tr.degKK(

n

• j=1

Θ(r1, . . . , rp)ηj) for all sufficiently large (r1, . . . , rp) ∈ Np (i. e., there exist s1, . . . , sp ∈ N such that the last equality holds for all elements (r1, . . . , rp) ∈ Np with r1 ≥ s1, . . . , rp ≥ sp); (ii) degtiΦη ≤ mi (1 ≤ i ≤ p), so that deg Φη ≤ m and the polynomial Φη(t1, . . . , tp) can be represented as Φη =

m1

• i1=0

. . .

mp

• ip=0

ai1...ip t1 + i1 i1

• . . .

tp + ip ip

• where ai1...ip ∈ Z for all i1, . . . , ip.

28

• Φη(t1, . . . , tp) is called the differential di-

mension polynomial of the extension L/K as- sociated with the set of differential generators η (and the given partition of the basic set ∆).

• For any permutation (j1, . . . , jp) of the set

{1, . . . , p}, we define the lexicographic order <j1,...,jp on Np as follows: (r1, . . . , rp) <j1,...,jp (s1, . . . , sp) if and only if either rj1 < sj1 or there exists k ∈ N, 1 ≤ k ≤ p − 1, such that rjν = sjν for ν = 1, . . . , k and rjk+1 < sjk+1. If Σ ⊆ Np, then Σ′ denotes the set {e ∈ Σ|e is a maximal element of Σ with respect to one of the p! lexicographic orders <j1,...,jp}. For example, if Σ = {(3, 0, 2), (2, 1, 1), (0, 1, 4), (1, 0, 3), (1, 1, 6), (3, 1, 0), (1, 2, 0)} ⊆ N3, then Σ′ = {(3, 0, 2), (3, 1, 0), (1, 1, 6), (1, 2, 0)}.

29

Theorem 8. Let K be a differential field whose basic set of derivations ∆ is a union of p disjoint finite sets (p ≥ 1): ∆ = ∆1 · · · ∆p, where ∆i = {δi1, . . . , δimi} (i = 1, . . . , p). Let L = Kη1, . . . , ηn and Φη =

m1

• i1=0

. . .

mp

• ip=0

ai1...ip t1 + i1 i1

• . . .

tp + ip ip

• the corresponding dimension polynomial.

Let Eη = {(i1, . . . , ip) ∈ Np|0 ≤ ik ≤ mk for k = 1, . . . , p and ai1...ip = 0}. Then d = deg Φη, am1...mp, elements (j1, . . . , jp) ∈ E′

η, the corresponding coeffi-

cients aj1...jp, and the coefficients of the terms

• f total degree d do not depend on η.

Furthermore, am1,...,mp = ∆-tr.degKL.

30