Some notes on basics in differential Galois theory

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Feb 5, 2007 - ferential Galois theory for Picard-Vessiot extensions”, which establishes the ...... ring with derivatio
Some notes on basics in differential Galois theory Benno Fuchssteiner∗ Kai Gehrs† February 5, 2007

Abstract. In this paper we summarize well-known results from differential Galois theory.

Contents 1 Preleminaries

2

2 Basic definitions and results 2.1 The fundamental theorem of differential Galois theory . . . . .

2 5

3 Second order linear differential equations 7 3.1 Some remarks on algorithms for finding rational, algebraic, exponential and Liouvillian solutions . . . . . . . . . . . . . . 9 3.2 Differential Galois groups of second order homogeneous linear differential equations . . . . . . . . . . . . . . . . . . . . . . . 11 4 Lie symmetries and differential Galois theory 14 4.1 Computing symmetric powers of linear differential operators . 16 5 On the use of symmetric powers of linear differential operators 18 ∗ †

Obernheideweg 19, D 33106 Paderborn, Germany University of Paderborn, Germany

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Literaturverzeichnis

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23

Preleminaries

In this paper we summarize well-known results of differential Galois theory. We start with some basic definitions finally leading us to the definition of Picard-Vessiot extensions. Some useful properties of this special class of differential field extensions are stated. In the next step we give a version of the “fundamental theorem of differential Galois theory for Picard-Vessiot extensions”, which establishes the well-known bijective correspondence between intermediate differential field extensions and subgroups of the differential Galois group of a Picard-Vessiot differential field extension. The presentation of these basic definitions and results in differential Galois theory mainly follows the book [4]. Afterwards, the relevant classes of solutions of second order homogeneous linear differential are defined. A well-known characterization of the differential Galois groups appearing among such equations is stated. Finally, we shortly present some examples of differential Galois groups of second order homogeneous linear differential equations. This part mainly follows the article [3]. Then we give a very brief review on the article [5], where some results from the field of Lie symmetries are compared to results from differential Galois theory for homogeneous linear differential euqations. Therefore the notion of symmetric powers of linear differential operators is introduced and a characterization of Lie symmetries is given. Further results emphasizing the importance of symmetric powers of linear differential operators in the field of differential Galois theory are summarized in the following section.

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Basic definitions and results

In the following we summarize some basic definitions, which are of importance in differential Galois theory. The following definitions and results are taken from the book [4]. Proofs are omitted, but there is always a reference where to find them in the literature. In the following, all fields are assumed to be of characteristic zero.

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Definition 2.1. (differential rings, ideals, fields and homomorphisms) (i) Let R be a commutative ring, 1R 6= 0R , and DR a derivation, i.e. DR : R → R is an additive map, such that DR (ab) = DR (a)b + aDR (b) for all elements a, b of the field R. Then the pair (R, DR ) is called a differential ring. (ii) Let R be a differential ring and DR its derivation. An ideal I ⊆ R is called a differential ideal, if D(I) ⊆ I. (iii) Let F be a field and DF a derivation, i.e. DF : F → F is an additive map, such that DF (ab) = DF (a)b + aDF (b) for all elements a, b of the field F . Then the pair (F, DF ) is called a differential field. (iv) Let R and S be differential rings with derivations DR and DS , respectively. A ring homomorphism f : R → S is called a differential homomorphism, if it commutes with the derivations, i.e. if DS (f (r)) = f (DR (r)) for all r ∈ R. If R and S are differential fields and f is a field homomorphism commuting with the derivations, then f is called a differential homomorphism of differential fields or for short differential homomorphism. (v) Let F be a differential field. An invertible differential homomorphism f : F → F is called a differential automorphism. We simply call a field F a differential field, if the derivation on F is clear. In this case we may also write D for reasons of abbreviation instead of DF , if it is clear, which derivation is meant. For ai ∈ F , 0 ≤ i ≤ n − 1, n ∈ N, n ≥ 2, we denote by L the linear differential operator L = Y (n) + an−1 Y (n−1) + . . . a1 Y (1) + a0 Y (0) ,

(2.1)

where Y is a variable and Y (i) = Di (Y ) denotes the i-th derivative of Y . Definition 2.2. (differential field extension) Let F be a differential field with derivation DF : F → F . A differential field E with derivation DE : E → E is called a differential field extension of F , if E ⊇ F and the restriction of DE to F equals DF . Proposition 2.3. (existence of differential field extensions for L = 0) Given a linear differential operator L with coefficients in the differential field F , there is always a differential field extension E ⊇ F , such that L = 0 has n independent solutions in F . Then E is called a differential field extension of F for L. 3

Proof. See [4], page ix of the Outline of Approach. Remark 2.4. (i) The field F in the proposition plays the role of a “splitting field” of the linear differential equation: as is classic Galois theory, where one deals with the set of solutions of univariate polynomial equations, there is always a field extension containing the roots of the equation under consideration. (ii) The proof of Proposition 2.3 is constructive. The differential field extension of E is defined as the fraction field Q(R), where R = F [yi,j ] is a polynomial ring in the n2 variables yi,j , 1 ≤ i ≤ n, 0 ≤ j ≤ n − 1, and yi,0 can be associated with the i-th solution yi of the n linearly independent solutions of L = 0 and yi,j with its j-th derivative Dj (yi ). Definition 2.5. (constants of a differential field) Let F be a differential field with derivation DF . Then a ∈ F with DF (a) = 0 is called a constant of F . The subfield of constants of a differential field F will be denoted by CF or simply C, if the underlying differential field is clear. Definition 2.6. (Picard-Vessiot extension) E ⊇ F is called a PicardVessiot extension of F for L if (i) E is generated over F as a differential field by solutions of L = 0 in E. (ii) The constants of E are the constants of F . (iii) L = 0 has n solutions in E linearly independent over the constants. Note, that (i) means that E is the smallest differential field extension of F containing the solutions of L = 0 mentioned in (iii). Theorem 2.7. (construction of Picard-Vessiot extensions) Let R be the polynomial ring as in Remark 2.4 (ii). Let   y1,0 y2,0 · · · yn,0  y1,1 y2,1 · · · yn,1    w = w(y1 , y2 , . . . , yn ) = det  .. (2.2)  .. ..  .  . . y1,n−1 y2,n−1 · · · yn,n−1 and define S = R[w−1 ]. Let P ⊆ S be a maximal prime differential ideal. Then the fraction field E = Q(S/P ) is a Picard-Vessiot extension of F for L. 4

Proof. The assertion follows from the results presented in [4] on pages ix and x of the Outline of Approach. Theorem 2.8. (uniqueness of Picard-Vessiot extensions) Any two Picard-Vessiot extensions of the differential field F for L are isomorphic over F. Proof. See [4], page xi of the Outline of Approach and pages 29 and 30, proof of Theorem 3.13. Definition 2.9. (group of differential automorphisms) Let F be a differential field and E ⊇ F a differential field extension of F . The group (with respect to composition) of differential automorphisms E → E, whose restriction to F is the identity map, is denoted by G(E/F ), i.e.   σ is a differential automorphism and G(E/F ) = σ : E → E . σ(a) = a for all elements a ∈ F If E is a Picard-Vessiot extension of F for L, then G(E/F ) is also referred to as the differential Galois group of L = 0.

2.1

The fundamental theorem of differential Galois theory

Again, as in the preceding section, the outline of theory is based on [4]. Proofs are again omitted, but again references to the proofs of the results are stated, such that it is easy to find them in the literature. Let F be a differential field and E a Picard-Vessiot extension of F for L = 0, L as defined in (2.1). In the following, we assume, that the subfield C of constants of F is algebraically closed. The fundamental theorem of differential Galois theory establishes a bijective correspondence between the intermediate differential fields K, E ⊇ K ⊇ F , and the subgroups H of the automorphism group G(E/K). Recall, that a subgroup H of G(E/F ) is a normal subgroup of G(E/F ) if σHσ −1 = H for all σ ∈ G(E/F ). In other words: H is a normal subgroup of G(E/F ) if and only if all elements of G(E/F ) commute with the elements of H. 5

Definition 2.10. Let G be a group of differential automorphisms F → F , F a differential field. Then we denote by F G = {a ∈ F | σ(a) = a ∀σ ∈ G} the differential subfield of F , whose elements are left invariant under all automorphism in G. Remark 2.11. With the above definitions, one can give an alternative characterization of Picard-Vessiot extensions. Let E ⊇ F be a differential field extension of the differential field F . Assume, that the subfield C of constants of F is algebraically closed. Then E is a Picard-Vessiot extension of F if and only if the following three conditions hold: (i) E = F hV i, where V ⊆ E is a finite-dimensional vector space over C. (ii) There is a group G of differential automorphisms of E with G(V ) ⊆ V and E G = F . (iii) E ⊇ F has no new constants, i.e. CE = C. If these three conditions hold and if y1 , . . . , yn is a C-basis of the finitedimensional vector space V over the field of constants C, then E is the Picard-Vessiot extension of F for the linear differential operator L=

w(Y, y1 , . . . , yn ) , w(y1 , . . . , yn )

(2.3)

where     w(Y, y1 , . . . , yn ) = det   

Y y1 D(Y ) D(y1 ) D2 (Y ) D2 (y1 ) .. .. . . n n D (Y ) D (y1 )

··· ··· ···

yn D(yn ) D2 (yn ) .. .

· · · Dn (yn )

      

and     w(y1 , . . . , yn ) = det   

y1 D(y1 ) D2 (y1 ) .. .

y2 D(y2 ) D2 (y2 ) .. .

··· ··· ···

yn D(yn ) D2 (yn ) .. .

Dn−1 (y1 ) Dn−1 (y2 ) · · · Dn−1 (yn ) 6

      

and L−1 (0) = V . For a proof of this characterization of Picard-Vessiot extensions see [4], pages 27 and 28, proof of Proposition 3.9. Remark 2.12. More precisely, the following version of the fundamental theorem of differential Galois theory is known as the “fundamental theorem of differential Galois theory for Picard-Vessiot extensions”. Theorem 2.13. (fundamental theorem of differential Galois theory) Let F be a differential field with algebraically closed field of constants C and E ⊇ F a Picard-Vessiot extension of F . Then there is a bijective correspondence between the set of intermediate differential fields F := {E ⊇ K ⊇ F | K is an intermediate differential field} and the set of subgroups G := {H ≤ G(E/F ) | H is a Zariski closed subgroup} of G(E/F ) given by the maps Φ : F → G, K 7→ G(E/K) and

Ψ : G → F, H 7→ E H ,

i.e. Φ ◦ Ψ = idG and Ψ ◦ Φ = idF . The intermediate field K is a PicardVessiot extension of F if and only if H = G(E/K) is normal in G(E/F ). If the latter is the case, then G(E H /F ) = G(E/F )/H. Proof. For a proof, see [4], Chapter 6, pp. 75, Propositions 6.1, 6.2, 6.3, 6.4 and their proofs as well as Theorem 6.5.

3

Second order linear differential equations

We now deal with second order homogeneous linear differential equations, i.e. equations of the form y 00 (x) + a(x)y 0 (x) + b(x)y(x) = 0,

(3.1)

where a(x) and b(x) are elements of the differential field F = C(x), i.e. a(x) and b(x) are rational complex functions. It is well known that the transformation  1Z x  a(τ ) dτ (3.2) y(x) → y(x) exp − 2 7

transforms (3.1) to the more special form y 00 (x) = r(x) · y(x),

(3.3)

where r(x) ∈ F (see for example [2], page 22). Since any second order homogeneous linear differential equation (3.1) can be transformed to an equation of type (3.3), we restrict our observations to equations of the latter type. The following classes of solutions are of interest: Definition 3.1. (rational, algebraic, primitive, exponential and Liouvillian solutions) Let η(x) ∈ F be a solution of (3.3): (i) η(x) is called rational, if η(x) ∈ F . (ii) η(x) is called algebraic, if it is the solution of a polynomial equation over F . (iii) η(x) is called primitive, if its derivative η 0 (x) is an element of F , i.e. Z x f (τ ) dτ η(x) = for some f (x) ∈ F . (iv) η(x) is called exponential or exponential of a primitive solution, if is an element of F , i.e. Z x  f (τ ) dτ η(x) = exp

η 0 (x) η(x)

for some f (x) ∈ F . (v) η(x) is called Liouvillian if there is a tower of differential fields F = E0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Em−1 ⊂ Em = E, such that η(x) ∈ E and for each i ∈ {1, . . . , m} we have Ei = Ei−1 (ηi (x)), where ηi (x) is either algebraic, primitive or exponential over Ei−1 and ηm (x) = η(x).

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Remark 3.2. (i) In the case of second order homogeneous linear differential equations, one obtains: If equation (3.3) has one Liouvillian solution, then all solutions of (3.3) are Liouvillian (see [3], Proposition 2.3.). (ii) Definition 3.1 (ii) implies, that for an algebraic solution η(x) of (3.3) one can find a minimal polynomial, i.e. a monic univariate polynomial of minimal degree µη (X) = a0 (x)+a1 (x)X +a2 (x)X 2 +. . .+al−1 (x)X l−1 +X l , ai (x) ∈ F for 0 ≤ i ≤ l, such that µη (η(x)) = 0. In [2] the author shows, that under special assumptions, these minimal polynomials can be expressed in terms of special differential invariants1 .

3.1

Some remarks on algorithms for finding rational, algebraic, exponential and Liouvillian solutions

Let F = C(x) and consider the linear differential operator L = Dn + an−1 Dn−1 + . . . a1 D + a0

(3.4)

over F , i.e. ai ∈ F for 0 ≤ i ≤ n − 1. First of all, we give the following statement from [6] (see page 10): Remark 3.3. Let b ∈ F and consider the inhomogeneous linear ODE L(y) = b. Define 0 1 L(y) . (3.5) Lh (y) := b b Then the following hold: (a) Any solution in F of L(y) = b is a solution of Lh (y) = 0. (b) For any solution v of Lh (y) = 0 there is a constant c such that v is a solution L(y) = c b. This construction allows us to reduce the solution of inhomogeneous n-th order linear ODEs to the solution of homogeneous (n+1)-st order linear ODEs. Hence, one can restrict attention to the homogeneous situation L(y) = 0. 1

In [2] the author gives explicit formulas for the computation of such minimal polynomials in terms of invariants according to a classification of the differential Galois groups of a homogeneous linear differential equations of a fixed order. See section 3.1 of [2] for details.

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Rational solutions of L(y) = 0. We search for solutions y of L(y) = 0, which are elements of F . Because y is a rational function, we can consider the formal Laurent series expansion of y at 0, i.e. X y= aα,m xm = aα,α xα + aα,α+1 xα+1 + aα,α+2 xα+2 + . . . . (3.6) m≥α

We search for the coefficients aα,m in this series expansion. In the same way we can consider Laurent series expansions of the functions a0 , a1 , . . . , an−1 , which form the coefficients of the linear differential operator L, i.e. X ai = aαi ,m xm (3.7) m≥αi

for 0 ≤ i ≤ n−1. The basic idea in determining y is then to insert the formal Laurent series expansion of y and the given Laurent series expansions of ai for 0 ≤ i ≤ n − 1 into the differential equation L(y) = 0 and to determine the searched for values for aα,m , m = α, α+1, α+2, . . . successively by comparing coefficients of the Laurent series expansions. Algebraic solutions of L(y) = 0. We search for solutions y of L(y) = 0, which can be expressed as solutions of polynomial equations with coeffcients in F , i.e. which occur as solutions of so–called minimal polynomials. For every algebraic solution y of L(y) = 0 there is a minimal polynomial P (z) ∈ F [z] such that P (y) = 0. Due to group theoretical classifications, it is possible to identify such minimal polynomials with the help of the differential Galois group of L(y) = 0. Such classifications have been done e.g. in [2]. Exponential solutions of L(y) = 0. We search for solutions y of L(y) = 0 0 such that yy ∈ F . The strategy to find such solutions is to study the associated Riccati ODE for L(y) = 0. In the case, where L is a second order differential operator and the associated differential Galois group is unimodular, the associated Riccati ODE for y 00 = a y, a ∈ F , is y 0 + y 2 = a. To determine the exponential solutions of y 00 = a y one can look at the rational solutions of y 0 + y 2 = a, since whenever g is a solution of y 00 = a y, then R g0 f := g is a solution of y 0 + y 2 = a, i.e. g = exp( f dx) is obviously an exponential solution of y 00 = a y. For computing rational solutions of Riccati ODEs Van der Put and Singer discuss methods in [6] (see Section page 106), which also base on the idea of using Laurent series expansions (similar to 10

the case we mentioned above in connection with the computation of rational solutions of L(y) = 0). Liouvillian solutions of L(y) = 0. Whenever L(y) = 0 has a non–zero Liouvillian solution, there always is at least one Liouvillian solution f such 0 that ff is an algebraic solution of the Riccati ODE associated with L(y) = 0. Group theoretical arguments allow to give a classification of the minimal polynomials of rational solutions of the Riccati ODE, which are of the form f0 . f

3.2

Differential Galois groups of second order homogeneous linear differential equations

Let ξ(x), η(x) ∈ E, E a Picard-Vessiot extension of F = C(x), be independent solutions2 of the second order homogeneous linear differential equation (3.3). Furthermore, let σ ∈ G(E/F ) be a differential automorphism contained in the differential Galois group of (3.3). Since σ maps solutions of (3.3) to solutions, we have   σ η(x) ξ(x) = η(x) ξ(x) · Mσ , (3.8) where Mσ is an invertible matrix of the form   aσ b σ ∈ C2×2 . Mσ = cσ dσ

(3.9)

This means Mσ ∈ GLC (2)3 . Furthermore, since the components of Mσ are constants, we obtain     η(x) ξ(x) η(x) ξ(x) σ = · Mσ . (3.10) η 0 (x) ξ 0 (x) η 0 (x) ξ 0 (x) Taking determinants provides σ(w(η(x), ξ(x))) = w(η(x), ξ(x)) · det(Mσ ), where again we denote by w(η(x), ξ(x)) the Wronskian of η(x), ξ(x), i.e.   η(x) ξ(x) w(η(x), ξ(x)) = det . (3.11) η 0 (x) ξ 0 (x) 2

We consider E as a C-vector space. Then ξ(x), η(x) are assumed to be linearly independent over field of constants C of F = C(x). 3 GLC (2) denotes the group of invertible (2 × 2)-matrices over the complex numbers.

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Since w(η(x), ξ(x)) ∈ C, we have σ(w(η(x), ξ(x))) = w(η(x), ξ(x)) and it follows det(Mσ ) = 1. This proves Mσ ∈ SLC (2)4 . We summarize our results: Theorem 3.4. (characterization of Galois groups of second order equations) The differential Galois group of a second order homogeneous linear differential equation can be identified with an algebraic subgroup of SLC (2). Now we state two examples, where we explicitly compute the differential Galois group of a second order homogeneous linear differential equation. The examples are taken from [3], pp. 6,7. Example 3.5. (differential Galois group of y 00 = y) Consider the second order homogeneous linear differential equation y 00 (x) = y(x) over F = C(x). A fundamental system of solutions is {exp(x), exp(−x)}. Hence, a PicardVessiot extension for the given ODE is E = F (exp(x)) ⊇ F . We want to compute the differential Galois group G(E/F ). For every σ ∈ G(E/F ) we obtain (σ(exp(x)))0 = σ(exp(x)0 ) = σ(exp(x)). It follows σ(exp(x)) = dσ exp(x) and, hence, σ(exp(−x)) = σ(exp(x)−1 ) = σ(exp(x))−1 = d−1 σ exp(−x), where dσ ∈ C. Thus, we obtain     dσ 0 σ exp(x) exp(−x) = exp(x) exp(−x) . 0 d−1 σ The differential Galois group G(E/F ) can be identified with the set of matrices    d 0 ; d ∈ C \ {0} ⊂ SLC (2). 0 d−1 Example 3.6. (differential Galois group of y 00 = − 4x12 y) Consider the second order homogeneous √ linear differential√equation y 00 (x) = − 4x12 y(x) over F = C(x). Define η(x) = x and ζ(x) = x ln(x). Then {η(x), ζ(x)} is a fundamental set of solutions of the differential equation. Hence, a PicardVessiot extension for the given ODE is E = F (η(x), ζ(x)) ⊇ F . We want to compute the differential Galois group G(E/F ). Let σ ∈ G(E/F ). Since σ maps solutions to solutions and E can be viewed as a vector space over F with basis {η(x), ζ(x)}, we have √ √ σ(η(x)) = c1 x + c2 x ln(x) 4

SLC (2) denotes the subgroup of (2 × 2)-matrices of GLC (2) having determinant 1.

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for some constants c1 , c2 ∈ C, and, hence,  √x ln(x)  1 + √ . σ(η(x)) = c1 √ + c2 x 2 x 2 x 0

(3.12)

But on the other hand, since σ is a differential automorphism, it commutes with the derivation, i.e.  1  1 √ σ(η(x))0 = σ(η 0 (x)) = σ √ = σ( x)−1 , 2 2 x i.e. σ(η(x))0 =

1 1 √ √ . 2 c1 x + c2 x ln(x)

(3.13)

Equating the right-hand-sides of (3.12) and (3.13) and afterwards expanding the products, provides c21 c2 1 + c1 c2 ln(x) + c1 c2 + 2 ln(x)2 + c22 ln(x) − = 0. 2 2 2 Since this must hold for any valid x, it follows c2 = 0 and c1 = ±1. Hence, we have proved, that σ(η(x)) = ±η(x).

(3.14)

Finally, we need to compute the image of ζ(x) under σ. Since σ leaves the elements of F invariant and commutes with the derivation, we find 1 1 σ(ln(x))0 = σ(ln(x)0 ) = σ = , x x i.e. σ(ln(x)) = ln(x) + cσ , cσ ∈ C the constant of integration. Hence, √ √ (3.15) σ(ζ(x)) = σ( x) σ(ln(x)) = ± x (ln(x) + cσ ) = cσ η(x) + ζ(x). From (3.14) and (3.15) we can conclude, that we can identify the differential Galois group G(E/F ) with set of matrices       1 c −1 c ;c∈C ∪ ;c∈C , 0 1 0 −1 which is again a subgroup of SLC (2). 13

4

Lie symmetries and differential Galois theory

W. R. Oudshoorn and M. Van der Put investigate in their article [5] possible connections between the theory of Lie symmetry analysis and differential Galois theory. They come to the conclusion, that there is no direct relation between these two fields. We give a short summary of the main results of [5]. First of all, the theory of Lie symmetries for linear homogeneous differential equations is developed from a purely algebraic point of view. We skip the sequence of formal definitions to define generators of Lie point symmetries here, since, to our mind, the definition itself is the same, which is well-known from the “jet bundle formalism”. The generators of Lie symmetries are written in the form of vector fields ∇ = ∇ξ,η = ξ(x, y)

∂ ∂ + η(x, y) ∂x ∂y

(4.1)

for some C ∞ -functions ξ(x, y) and η(x, y). This is a usual way of introducing Lie symmetries already known from the standard book [10] by H. Stephani. The Lie algebra of all such vector fields ∇ξ,η together with their prolongations to higher derivatives5 is denoted by L, where the commutator of two such vector fields ∇1 and ∇2 is defined by J∇1 , ∇2 K = ∇1 ∇2 − ∇2 ∇1 .

(4.2)

For a given differential equation ω(x, y(x), y 0 (x), . . . , y (n) (x)) = 0 polynomial in its arguments the algebra of Lie (point) symmetries Lω is defined to be the set of all elements ∇ ∈ L, such that ∇(ω) ⊆ hwi, where hwi denotes the differential ideal generated by ω. One of the central results presented in [5] in the above described setting, is Proposition 4.1. (Oudshoorn, Van der Put, 2001) The algebra of Lie symmetries of a second order homogeneous linear differential equation of the from y 00 (x) + a(x)y 0 (x) + b(x)y(x) = 0, a(x), b(x) ∈ F , F a differential field 5

For details on the definition of these prolongations and on formulas, how to compute them, see also [10].

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with algebraically closed field of constants C, is, after a possible extension of the field F , isomorphic to SLC (3). Proof. See [5], pp. 353-353, Proposition 3.1. Form this proposition and Theorem 3.4, where we saw, that the differential Galois group of a second order homogeneous linear differential equation over C(x) can always we identified with a subset of SLC (2) we can directly conclude: Corollary 4.2. If y 00 (x) + a(x)y 0 (x) + b(x)y(x) = 0 is a homogeneous linear differential equation with coefficients a(x), b(x) ∈ F = C(x), then the Lie algebra of Lie symmetries and the differential Galois group are not isomorphic. Nevertheless, there is one more interesting result, which establishes a connection between the Lie symmetries of the differential equations associated with the so-called symmetric powers of linear differential operators L = D2 + a1 (x)D + a0 (x) and the Lie symmetries of the differential equation given by L itself. Let {f1 , f2 } be a fundamental set of solutions of Ly = 0. The monic operator Ln associated with the homogeneous linear differential equation having {f1d f2e | 0 ≤ d, d + e = n} as a fundamental set of solutions, is called the n-th symmetric power of L. For these operators we find: Proposition 4.3. (Oudshoorn, Van der Put, 2001) Let L = D2 + a1 (x)D + a0 (x), a1 (x), a0 (x) ∈ F , F a differential field. (i) Ln has the same Lie symmetries as L. (ii) Ln is of the form Dn + bn a1 (x)Dn−1 + (cn a1 (x)2 + dn a01 (x) + en a0 (x))Dn−1 + . . . , where bn , cn , dn , en are the constants n(n − 1) , 2 n(n − 1)(n − 2) dn = , 2

n(n − 1)(n − 2)(3n − 1) , 24 (n + 1)n(n − 1) en = . 6

bn =

cn =

Proof. See [5], pp. 354-355, Proposition 3.2. 15

4.1

Computing symmetric powers of linear differential operators

In this seubsection we discuss some well-known results and algorithmic ideas to compute the n-th symmetric power Ln of a linear differential operator L. We start with the case, where L is a second order linear differential operator. Two methods to compute symmetric powers – one by Van der Put et. al. mentioned in [5] and one by Bronstein et. al. mentioned in [1] – are roughly sketched. At the end of the subsection, we present a theorem to compute the symmetric power of an n-th order linear differential operator in an arbitrary differential ring R with derivation D, where R is an integral domain of characteristic 0. This methods can also be found in [1]. Remark 4.4. The proof of 4.3 to be found in [5] is constructive. The authors sketch the following algorithm to compute the n-th symmetric power of L = D2 + a1 (x)D + a0 (x), a1 (x), a0 (x) ∈ F , F a differential field: • Let e0 be a non-zero solution of Ly = 0 and define monic operators Mi of degree i for i = 0, . . . , n − 1 by Mi en−1 = (n − 1) · . . . · (n − i) · e0n−i−1 · (e00 )i . 0 This provides M1 = D and Mi+1 = (D + ∗a1 )Mi + ∗a0 Mi−1 , where ∗ denotes some constants. • For certain integers ∗, the operator T := (D + ∗a1 )Mn−1 + ∗a0 Mn−2 maps en−1 to 0. 0 • It follows, that T = Ln , where the integers ∗ can be computed by choosing special cases for a0 and a1 . A more explicit version of an algorithm to compute symmetric powers of monic second order linear differential operators is presented in [1]. M. Bronstein et. al. even treat a more general situation, where the coefficients of the linear differential operator are elements of differential ring R, which is an integral domain of characteristic 0. In this situation, a special treatment for non-monic linear differential operators becomes necessary, since the leading coefficient might not be a unit of the ring R. Since we are only interested in the case, where the underlying coeffcient domains is a differential field, only Theorem 1 of [1] is of interest for us. This theorem presents an explicit iteration formula to compute the n-th symmetric power of a linear differential 16

operator L = D2 + a(x)D + b(x), where a(x), b(x) ∈ F , F some differential field. Theorem 4.5. (Bronstein, Mulders, Weil, 1997) Let L = D2 +a(x)D + b(x), where a(x), b(x) ∈ F , F some differential field and D the derivation in F . Let n > 0 and define M0 := 1, M1 := D and Mi+1 := (D + ia(x))Mi + i(n − (i − 1))b(x)Mi−1 . Then Mn+1 = Ln , i.e. Mn+1 is the n-th symmetric power of L. Proof. See [1], page 157. A more general method not restricted to differential fields and second order linear differential operators is obtained from the following theorem: Theorem 4.6. (Bronstein, Mulders, Weil, 1997) Let R be a differential ring with derivation D, R an integral domain of characteristic 0. Let L=

n X

ai (x) · Di ,

n > 0,

i=0

a linear differential operator, Y0 , . . . , Yn−1 be indeterminates and ∆ = an D a derivation on the polynomial Pn−1ring R[Y0 , . . . , Yn−1 ] with ∆Yi = an Yi+1 , 0 ≤ ai Yi . Let m > 0 and define i ≤ n − 2 and ∆Yn−1 = − i=0 ω0 = Y0m

and

wi+1 = ∆ωi − iD(an (x))ωi

for PMi ≤ 0. Then there is a minimal positive integer M ∈ N, such that i=0 ci (x)ωi = 0, ci (x) ∈ R, 0 ≤ i ≤ M , and not all ci (x) equal to zero. Then M X  ci (x)an (x)i Di (Y0m ) = 0, i=0

i.e. the m-th symmetric power Lm of L can be obtained from the first linear dependence over R between ω0 , ω1 , . . .. Proof. See [1], page 158.

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Remark 4.7. Theorem 4.6 implies, that the m-th symmetric power of L is given by M X Lm = ci (x)an (x)i Di . i=0

The authors of [1] remark, that finding the first linear dependence among the above defined ω0 , ω1 , . . . is equivalent to computing the kernel of a matrix over R. Nevertheless, the above described procedure produces increasing expression swell, which makes the symbolic computations difficult in praxis. The proof of Theorem 4.6 is done by induction in i to show, that ωi = ain Di (Y0m ). Then it follows directly, that M X i=0

i

ci (x)an (x) D

i



(Y0m )

=

M X

i

ci (x)an (x) D

i

(Y0m )

i=0

=

M X

ci (x)ωi = 0.

i=0

Remark 4.8. Note, that if R = F is a differential field in the situation of Theorem 4.6, we can assume an (x) = 1, i.e. we can assume, that the linear differential operator L is monic. The iteration formula then simplies to ω0 = Y0m

and wi+1 = ∆ωi

for i ≤ 0, since D(an (x)) = D(1) = 0.

5

On the use of symmetric powers of linear differential operators

In the preceding section we presented two algorithmic approaches to compute the symmetric powers of linear differential operators of order 2. In this section we give a short summary of results from the field of differential Galois theory related with the notion of symmetric powers of linear differential operators. For second and third order linear differential operators L the symmetric powers Lm , m ∈ N, as defined in the previous section, are used by Singer et. al. in [7] to characterize necessary and sufficient conditions for the existence of Liouvillian solutions of the homogeneous linear differential equation L(y) = 0. By inspecting factors of Lm , Singer et. al. present results, which 18

allow to determine the structure of the differential Galois group of L = 0 with the help of symmetric powers of L. Some of the central results of [7] for second order homogeneous linear differential equations are summarized in the following theorem. Theorem 5.1. (Singer & Ulmer, 1997) Let F be a differential field and L(y(x)) = y 00 (x) + r(x)y(x) = 0 a second order homogeneous linear differential equation6 over F with unimodular differential Galois group7 . (i) L(y(x)) = 0 has Liouvillian solutions if and only if the sixth symmetric power L6 of L is reducible over F . (ii) The Galois group of L(y(x)) = 0 is the Tetrahedral Group A3SL2 if and only if the second symmetric power L2 of L is irreducible over F and the third symmetric power L3 of L is reducible over F . (iii) The Galois group of L(y(x)) = 0 is S4SL2 if and only if the third symmetric power L3 of L is irreducible over F and the fourth symmetric power L4 of L is reducible over F . 2 (iv) The Galois group of L(y(x)) = 0 is ASL if and only if the fourth 5 symmetric power L4 of L is irreducible over F and the sixth symmetric power L6 of L is reducible over F .

(v) The Galois group of L(y(x)) = 0 is SL2 (C) if and only if the sixth symmetric power L6 of L is irreducible over F . 6 Note, that the special form of the homogeneous linear differential equation is not a restriction of the generality, since any homogeneous linear differenial equation can be R transformed to such a form via y = z exp(− 12 a1 , where a1 is the coefficient of y 0 in the original equation. 7 The fact, that the differential Galois group of L has been assumed to be unimodular can be viewed as a technical detail not restricting the generality of the theorem. By Theorem 3.2 of [7] or Theorem 1.2 of [8] one knows, that any homogeneous linear differential equation can be transformed to a homogeneous linear differential equation, such that the differential Galois group of this new equation is unimodular. The explicit transformation for a monic R an−1 linear differential equation is y = z · exp(− n ), where n is the order of the linear differential operator associated with the homogeneous linear differential equation and an−1 the coefficient of the (n − 1)-st power of D in the operator. In fact, if a second order homogeneous linear differential equation is of the form y 00 (x)+r(x)y(x) = 0, it’s differential Galois group is automatically a unimodular group.

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Proof. One can find the result in (i) in the framework of Proposition 4.4, [7]. The results in (ii), (iii), (iv) and (v) are part of Proposition 4.3, [7]. For third order homogenenous linear differential equations, we find the following results in [7]: Theorem 5.2. (Singer & Ulmer, 1997) Let F be a differential field and L(y(x)) = y 000 (x) + r(x)y 0 (x) + s(x)y(x) = 0 a third order homogeneous linear differential equation8 over F with unimodular differential Galois group9 . Then L(y(x)) = 0 is solvable in terms of lower order linear differential equations if and only if the fourth symmetric power L4 of L has order less than 15 or is reducible. Proof. For a proof of the theorem see [7], Corollary 4.8. Theorem 5.3. (Singer & Ulmer, 1997) Let L(y(x)) = 0 be an irreducible third order homogeneous linear differential equation with unimodular differential Galois group over some differential field F , where the field CF of constants of F is algebraically closed. L(y(x)) = 0 has a Liouvillian solution if and only if • the fourth symmetric power L4 of L has order less than 15 and • the second symmetric power L2 of L has order 6 and is irreducible or • the third symmetric power L3 of L has a factor of order 4. Proof. See [7], Corollary 4.9. Remark 5.4. A classification of the differential Galois groups of homogeneous third order linear differential, which are irreducible and have a unimodular differential Galois group, due to symmetric powers of the associated linear differential operator is given in the framework of Corollary 4.10 in [7]. Additionally, in section 5 of [7] examples of concrete homogeneous linear differential equations of order 2 and 3 are discussed. 8

Note, that the assumption of this special form is not a loss of generality. Any third order homogeneous linear differential equation can be transformed to such a form via R a2 y = z · exp(− 3 ), where a2 is the coefficient of y 00 (x) 9 In fact, if a third order homoegenous linear differential equation is of the form y 000 (x)+ r(x)y 0 (x)+s(x)y(x) = 0, its differential Galois group is automatically a unimodular group.

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In his paper [1], M. Bronstein states, that M. Singer and F. Ulmer prove in their work presented in [8], that the Liouvillian solutions of second and third order homogeneous linear differential equations are closely connected with the algebraic (radical) solutions of symmetric powers of the linear differential operator associated. Indeed, first of all, certain differential semi-invariants10 of degree m of homogenenous linear differential equations L(y(x)) = 0 with finite differential Galois group can be computed as non-trivial rational solutions of the (m · i)-th symmetric power Lm·i of L, where i divides the order of a dimensional character of the finite differential Galois group of L(y(x)) = 0. The main role of the symmetric powers in the context of the work presented in [8] is in fact, that the coefficients of the minimal polynomial of an algebraic logarithmic derivative of a solution of an irreducible homogeneous linear differential equation can be determined as solutions of symmetric powers11 . Ulmer et. al. note in their paper [11], that the differential invariants of degree m of the differential Galois group are rational solutions of the m-th symmetric power of the linear differential operator associated. In the case of second order homogeneous linear differential equations L(y(x)) = y 00 (x) + a1 (x)y 0 (x) + a0 (x) the authors investigate solutions of the associated Riccati differential equation given by Ri(u(x)) = u0 (x)−a0 (x)−a1 (x)u(x)+u(x)2 = 0. Furthermore, the authors of [11] state, that to compute Liouvillian solutions of L(y(x)) = 0, one can compute the minimal polynomial P (u) of an algebraic solution of Ri(u(x)) = 0. One of the main results presented in this context in [11] is: Theorem 5.5. (Ulmer, Weil, 1996) Let L(y(x)) = y 00 (x) + a1 (x)y(x) + a0 (x)y(x), ai (x) ∈ F , F some differential all zeroes of the minP field. Then i imal polynomial P (u(x)) = u(x)m + m−1 b (x)u(x) with bi (x) ∈ F are soi i=0 lutions of the associated Riccati equation Ri(u(x)) = 0 if and only if bm−1 (x) is the logarithmic derivative of an exponential solution (over F ) of the m-th 10

A possible definition of differential semi-invariants is the following: Let V a vector space over the field C with basis {y1 , . . . , yn } and G ⊆ SL(V ) a linear group. Define an action of g ∈ G on the multi-variate polynomial ring C[y1 , . . . , yn ] by g(p(y1 , . . . , yn )) := p(g(y1 ), . . . , g(yn )). A polynomial with the property that for all g ∈ G we have g(p(y1 , . . . , yn )) = ψp (g) · p(y1 , . . . , yn ) with ψp ∈ C, is called a semi-invariant of G. If for all g ∈ G we have ψp (g) = 1, then p(y1 , . . . , yn ) is called an invariant of G. See also [11], Definition 4. 11 Section 5 of [8] summarizes the algorithmic ideas to determine solutions out of minimal polynomials, which themselves are computed with the symmetric of symmetric powers.

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symmetric power Lm , i.e. bm−1 is a semi-invariant of the differential Galois group of L(y(x)) = 0. Proof. For a proof of the theorem see [11], proof of Theorem 5. Additionally, Ulmer et. al. present in [11] an algorithm, which uses rational solutions of m-th symmetric powers of second order linear differential operators to find Liouvillian solutions of second order homogeneous linear differential equations. This algorithm is presented and discussed in detail in the framework of section 3 of [11]. Depending on the fact, whether the differential Galois group of a second order homogeneous linear differential equation is reducible, primitive or imprimitive12 , the authors show, that the computation of Liouvillian solutions corresponds to the computation of rational solutions of symmetric powers13 . Finally, to close this section on symmetric powers, we refer to the paper [9] by Singer et. al. Here, algorithmic ideas to compute Liouvillian solutions of n-th order homogeneous linear differential equations are discussed. Again, the computation of coeffcients of minimal polynomials of solutions is mentioned to be done with the help of symmetric powers14 . Nevertheless, the authors mention, that in the computation of symmetric powers of higher order linear differential operators a huge expression swell is involved. Hence, Singer et. al. also discuss alternative ways in which the number of computations of symmetric powers is reduced and replaced by certain factoring routines for semi-invariant forms. Finally, it remains to state, that the results in [2], where symmetric powers of linear differential operators are involved, mostly base on the papers [7], [8], [9] by Singer et. al. and [11] by Ulmer et. al. 12

For explicit definitions of this group theoretical properties see for [2], Kapitel 2, Abschnitt 2.1, or [7], Section 2 on group theory. 13 For a detailed version of these critiria see Lemma 12, Lemma 13 and Lemma 14 and their proofs in section 3.1, 3.2 and 3.3 of [11]. 14 See section 3 on Bounds on the degree of the forms in [9].

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References [1] M. Bronstein, T. Mulders, J.-A. Weil: On Symmetric Powers of Differential Operators, ACM Press, Proceedings of the 1997 international symposium on Symbolic and algebraic computation, 1997 [2] W. Fakler: Algebraische Algorithmen zur L¨osung von linearen Differenzialgleichungen, B. G. Teubner Stuttgart Leipzig, MuPAD Reports, 1999 [3] J. J. Kovacic: An algorithm for solving second order linear homogeneous differential equations, Notes on a talk at the City College of The City University of New York, http://mysite.verizon.net/jkovacic/, 2005 [4] A. R. Magid: Lectures on Differential Galois Theory, American Mathematical Society, University Lecture Series Volumne 7, 1994 [5] W. R. Oudshoorn, M. Van der Put: Lie Symmetries And Differential Galois Groups of Linear Equations, Mathematics of Computation, Volumne 71, Number 237, pp. 349-361, 2001 [6] M. Van der Put, M. F. Singer: Galois Theory of Linear Differential Equations, Springer Verlag, Berlin · Heidelberg · New York · Hong Kong · London · Milan · Paris · Tokyo, Grundlehren der mathematischen Wissenschaften (A series of comprehensive studies in mathematics), Volume 328, 2003 [7] M. Singer, F. Ulmer: Galois Groups for second and third order linear differential equations, Journal of Symbolic Computation 11 (1997), pp. 136 [8] M. Singer, F. Ulmer: Liouvillian and algebraic solutions of second and third order linear differential equations, 11 (1997), pp. 37-73 [9] M. Singer, F. Ulmer: Linear differential equations and products of linear forms, Journal of Pure and Applied Algebra, 117-118, 1997, pp. 549-563 [10] H. Stephani: Differential Equations . Their solutions using symmetries, Cambridge University Press, Cambridge New York Port-Chester Melbourne Sidney, 1989 [11] F. Ulmer, J.-A. Weil: Note on Kovacic’s algorithm, Journal of Symbolic Computation 22 (1996), pp. 179-200 23