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 ∆ = { δ 1 , . . . , δ m } or an inversive difference field with basic set of automorphisms σ = { α 1 , . . . , α m } . 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 ∗ = { α 1 , . . . , α m , 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 = δ k 1 1 . . . δ k m ∈ Θ m m � ( k i ∈ N ) as ord θ = k i and set i =1 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 of an element γ = α l 1 1 . . . α l m m ∈ Γ ( l i ∈ Z ) is m � defined as ord γ = | l i | and for any r ∈ N , i =1 we set Γ( r ) = { γ ∈ Γ | ord γ ≤ r } . 3
• Let L = K � η 1 , . . . , η n � be a ∆- or σ ∗ - field extension of K generated by a finite set η = { η 1 , . . . , η n } . As a field, L = K ( { θη j | θ ∈ Θ , 1 ≤ j ≤ n } ) in the differential case and L = K ( { γη j | 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 ω η | K ( t ) ∈ Q [ t ] such that (i) ω η | K ( r ) = tr.deg K K ( { θη j | θ ∈ Θ( r ) , 1 ≤ j ≤ n } ) for all sufficiently large r ∈ Z ; (ii) deg ω η | K ≤ m and ω η | K ( t ) can be written m � t + i � � as ω η | K ( t ) = a i where a i ∈ Z . i i =0 (iii) d = deg ω η | K , a m and a d do not depend on the set of differential generators η of L/K ( a d � = a m iff d < m ). Moreover, a m is equal to the differential (∆-) transcendence degree of L over K (denoted by ∆- tr.deg K L ), that is, to the maximal number of elements ξ 1 , . . . , ξ k ∈ L such that the family { θξ i | θ ∈ Θ , 1 ≤ i ≤ k } is algebraically independent over K . 5
• The polynomial ω η | K ( t ) is called the differ- ential dimension polynomial of the ∆-field extension L/K associated with the set of ∆- generators η = { η 1 , . . . , η n } . • The numbers d = deg ω η | K and a d are called the differential (or ∆-) type and typi- cal differential (or ∆-) transcendence degree of the extension L/K ; they are denoted by ∆- type K L and ∆- t.tr.deg K L , 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 { y 1 , . . . , y n } be the algebra of differential (∆-) polynomials over the ∆-field K . (Recall that R is the polynomial ring K [ { θy j | θ ∈ Θ , 1 ≤ j ≤ n } ] in a denumer- able set of indeterminates θy j treated as a dif- ferential ring extension of K where δ ( θy j ) = ( δθ ) y j .) By a system of algebraic differential equa- tions over K we mean a system of the form f i ( y 1 , . . . , y n ) = 0 ( i ∈ I ) where { f i } i ∈ I ⊆ R ; by a solution we mean an n -tuple with coordinates in some differential field extension of K that annuls all f i . 7
Let P be the differential ideal generated by { f i | i ∈ I } in R . If it is prime, then Q ( R/ P ) = K � η 1 , . . . , η n � where η j is the image of y j in R/ P . By Theorem 1, we obtain a numerical polynomial ω η | 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 of 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, ∆ = { δ 1 , . . . , δ m } ) and let L = K � η 1 , . . . , η n � be a ∆-field extension of K . Let F be an intermediate differential field of the extension L/K and for any r ∈ N , let F r = F � K ( { θη j | θ ∈ Θ( r ) , 1 ≤ j ≤ n } ). Then there exists a polynomial ω K,F,η ( t ) ∈ Q [ t ] such that (i) ω K,F,η ( r ) = tr.deg K F r for all sufficiently large r ∈ Z ; (ii) deg ω K,F,η ≤ m and ω K,F,η ( t ) can be written as m � t + i � � ω K,F,η ( t ) = b i where b i ∈ Z . i i =0 (iii) d = deg ω K,F,η ( t ), b m and b d do not depend on the set of ∆-generators η of L/K . Furthermore, b m = ∆- tr.deg K F . 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 δ i , is expressed by a polynomial function. (We mean that δ i G = Gδ i 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 ω K,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.deg K ( E � K ( { θη j | θ ∈ Θ( r ) , 1 ≤ j ≤ n } )) Indeed, let K be an ordinary differential field with one basic derivation δ , let L = K � y � be the differential field of fractions of one differ- ential indeterminate y over K , and let E = K ( δ 2 y, δ 4 y, . . . , δ 2 k y, . . . ). Then � tr.deg K ( E K ( { θy | θ ∈ Θ( r ) } )) = [log 2 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 = { α 1 , . . . , α m } , Char K = 0) and let L = K � η 1 , . . . , η n � . Then there exists a polynomial φ η | K ( t ) ∈ Q [ t ] with the following properties. (i) φ η | K ( r ) = tr.deg K K ( { γη j | γ ∈ Γ( r ) , 1 ≤ j ≤ n } ) for all sufficiently large r ∈ N . (ii) deg φ η | K ( t ) ≤ m and the polynomial φ η | K ( t ) can be written as m � t + i � a i 2 i � φ η | K ( t ) = where a i ∈ Z . i i =0 (iii) The integers a m , d = deg φ η | K and a d do not depend on η . Also, a m = σ - tr.deg K L , the difference transcendence degree of L/K (this is the maximal number of elements ξ 1 , . . . , ξ k ∈ L such that the set { γξ i | γ ∈ Γ , 1 ≤ i ≤ k } is algebraically independent over K ). 13
The polynomial φ η | 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 φ η | K and the coefficient a d are called the inversive dif- ference (or σ ∗ -) type and typical inversive dif- ference (or typical σ ∗ -) transcendence degree of L over K . They are denoted by σ ∗ - type K L and σ ∗ - t.trdeg K L , respectively. Note that if the generators η 1 , . . . , η n are σ - algebraically independent over K and φ η | K ( t ) is the corresponding σ ∗ -dimension polynomial of L/K , then m � m �� t + k � ( − 1) m − k 2 k � φ η | K ( t ) = n . k k k =0 14
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