Computation of Affine Covariants of Differential Systems. 3. Let G be a linear group acting rationally on a finite-dimensional vector space W,. GL(W) the group of ...
COMPUTATION OF AFFINE COVARIANTS OF BIVARIATE DIFFERENTIAL SYSTEMS D. Boularas1 Abstract. This paper deals with affine covariants of autonomous differential systems. We give a constructive method for computing them. This method allows in particular to deduce a (minimal) system of generators of the algebra of affine covariants from one of centro-afffine invariants. In the case of planar quadratic differential systems, we give a minimal system of the algebra of center-affine covariants and using the previous method we construct a minimal system of generators of the algebra of affine covariants. Computations are made with Maple. All algorithms constitute the package SIB. Key words. Nonlinear differential systems, transformations of differential systems, classical invariant theory, classical groups, covariants. Subject classifications. 34A34, 34C20, 14L30, 14L35.
1. Motivations Let C be the real or complex field, V the C-vector space C n and A(n, m) the set of autonomous differential systems m n n ! dxj ! ! = ··· ajα1 α2 ...αk xα1 xα2 . . . xαk , j ∈ {1, . . . , n} dt α =1 k=0 α =1 1
(1.1)
k
where x = T (x1 , x2 , . . . , xn ) ∈ V is represented with superscript indices and coefficients ajα1 α2 ...αk and “time” t belong to C. The right hand of (1.1) is an n - tuple of polynomials of degree at most m. This paper is motivated by different applications of classical invariant theory in the study of differential systems (1.1) [10, 14] : for instance, the characterisation of each group orbit (normal forms, . . . ) or the description of particular trajectories (singularities, . . . ). LACO, D´epartement de Math´ematiques, Facult´e des Sciences, Universit´e de Limoges, 123, Avenue A. Thomas, 87000, Limoges, France 1
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D. Boularas
Many works [9, 10, 11] are devoted to invariants of differential systems with respect to the group of invertible matrices called the center-affine group. Systems of generators of invariants are constructed and used in qualitative study of differential systems (number and nature of critical points, first and particular algebraic integrals, symmetry axes of vector fields, . . . ). From the point of view of symbolic computation, this approach is important and fruitful because it allows us to express geometric properties of trajectories with the help of algebraic or semi-algebraic relations which depend on the coefficients of given systems. Furthermore covariants may be used to solve some problems that arise with numerical methods. The idea to use systematically the classical invariant theory in qualitative theory of differential systems is due to C.S. Sibirskii ([9, 10]). Up to now, few mathematicians have used the center-affine group of transformations of the phase space V . Our contribution is to propose a systematic method to construct systems of generators of affine covariants from those of center-affine invariants. This method is easy to implement; we have done it with Maple for planar quadratic systems. Other motivation of this work could be related to invariant theory. It is well-known that if a group is reductive, the algebra of invariants of all its representations are finitely generated. Few people try to determine the upper bound of degrees of generators of a given algebra of invariants. In ([1]), the authors give a complete and interesting description of this problem. However, up to now, there is no constructive method that gives a good bound ([1]). Only some algebraic forms and families of matrices give good examples of exact upper bound of generators. The two-dimensional quadratic systems give another example.
2. Introduction and notations The space of algebraic forms of degree k can be looked upon as the quotient space of V ∗⊗k by the subspace generated by all elements x1 ⊗ . . . ⊗ xi ⊗ xi+1 . . . ⊗ xk − x1 ⊗ · · · ⊗ xi+1 ⊗ xi · · · ⊗ xk . We denote Sk this space. The homogeneous part of degree k of the polynomial vector field is a n-tuple of elements of Sk . It should be identified with the vector space "(1.1) ∗ Sk V denoted by Sk1 . For example, S11 is the linear part of (1.1). In tensorial language, Sk1 is the space of tensors once contravariant and k times covariant which are symmetric with respect to the subscript indices. Consequently, 1 A(n, m) $ S01 ⊕ S11 ⊕ · · · ⊕ Sm .
Computation of Affine Covariants of Differential Systems
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Let G be a linear group acting rationally on a finite-dimensional vector space W,
GL(W) the group of automorphisms of W and
ρ : G &−→ GL(W) the corresponding rational representation. Let C[W] be the algebra of polynomials whose indeterminates are the coordinates of a generic vector of W. Definition 1. A polynomial function K ∈ C[W] is said to be a G - invariant of W if there exists a character λ of the group G such that ∀g ∈ G, K ◦ ρ(g) = λ(g).K. Here, The character of the group G is a rational (commutative) morphism of group G into Cm where Cm is the multiplicative group of C. If λ(g) ≡ 1, then the invariant is said absolute. Otherwise, it is said relative. In our situation, W should be A(n, m) or A(n, m) × V and the group G one of the following classical groups : 1. Gl(n) : group of center-affine transformations or invertible matrices, 2. T (n) : group of translations, 3. Aff(n) = T (n) ! Gl(n) : semi-direct group of affine transformations. In this work, the notion of covariant is taken in the following precise sense : Definition 2. A G - covariant of A(n, m) is a G - invariant of A(n, m) × V . For example, det(x, Ax, A2 x, . . . , An−1 x) where A is the linear part of (1.1) is a relative Gl(n) - covariant of A(n, m). Through this paper, we have a great deal with tensors. For sake of reasons of simplification of expressions we shall use the Einstein summation convention (for n ! n ! α1 j instance, aα1 α2 ajα2 stands for the same as ajα1 α2 aαjα12 ). j=1 α1 =1
Using this notations, the system (1.1) becomes dxj dt
= aj + ajα1 xα1 + ajα1 α2 xα1 xα2 + . . . + ajα1 α2 ...αm xα1 xα2 . . . xαm , j, α1 , α2 , . . . , αm ∈ {1, . . . , n}
(2.2)
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The transformations (xi ) &→ (Qij xj ), (xi ) &→ (xi − pi ), (xi ) &→ (Qij xj − pi ) of the above groups Gl(n), T (n) and Aff(n) transform each system (1.1) of A(n, m) into the system of A(n, m) defined by the formula : [ρ1 (P )(a)]jα1 α2 ...αk [ρ2 (p)(a)]jα1 α2 ...αk
= Qji Pαβ11 Pαβ22 . . . Pαβkk aiβ1 β2 ...βk , =
m−k !# i=0
$ k+i ajα1 α2 ...αk β1 β2 ...βi pβ1 pβ2 . . . pβi , i
(2.3) (2.4)
[ρ3 (P, p)(a)]jα1 α2 ...αk = m−k !# i=0
$ k+i Qji Pαγ11 Pαγ22 . . . Pαγkk ajγ1 γ2 ...γk β1 β2 ...βi pβ1 pβ2 . . . pβi , i
(2.5)
where P is the inverse # of the$matrix Q (or the linear part of g) of the corresponding (k + i)! k+i transformation and is the binomial coefficient . i k!i!
In definition 1, when G is one of the groups Gl(n), T (n), Aff(n), it is well-known ([2, 10]) that the character λ is equal to det(g)−κ where the integer −κ is called the weight of the invariant K. Clearly, every relative invariant of Gl(n) is an absolute invariant of Sl(n). The converse is also true (for homogeneous polynomials). This allows us to consider the set of Gl(n) - covariants (respectively invariants) of A(n, m) as an C - algebra, denoted by K(n, m) (respectively I(n, m) ). Similarly, the set of Aff(n) - covariants (respectively Aff(n) - invariants ) of A(n, m) is a C - algebra, denoted Q(n, m) (respectively J (n, m)). It is obvious that J (n, m) ⊂ I(n, m) and Q(n, m) ⊂ K(n, m). Let us return to the general case of a group G and a vector space W. Denote by C[W]G the algebra of Gl(n) - invariants of W. Following V.L. Popov [7], the main problem of the classical theory of invariants is to describe explicitly the algebras C[W]G . The idea of the description is as follows : 1. see whether C[W]G has a finite system of generators; 2. if it does, give a constructive method for finding a minimal system (ideally, find it exlicitly) of generators of C[W]G .
Computation of Affine Covariants of Differential Systems
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For the reductive groups G (like Gl(n) [1], this fact is known from Hilbert) the first problem is positively solved. However this is not the case for the affine group. The main results of this work consist in the following. After recalling some definitions and results about center-affine invariants ( 3) we construct effectively an isomorphism between the algebras I(n, m) and Q(n, m) ( 4). That means in particular that we are able to deduce a minimal system of generators of Q(n, m) from that one of I(n, m). This is the situation of I(2, 2). Of course the affine covariants are polynomials in center-affine covariants. In section 5 we achieve the construction of a minimal system of generators of K(2, 2). In the end ( 6) we give an effective computation with Maple of a minimal system of generators of Q(2, 2). The produced algorithms constitute the package SIB. This package contains the minimal systems of generators of I(n, m), K(n, m) and Q(n, m). It permits to express any center-affine or affine covariant with repect to the center-affine generators and computation for each multi-degree the corresponding syzygies. The affine covariants may be used in the study of normal forms and the qualitative behaviour of trajectories of systems (2.2) whose properties are invariant w.r.t. the affine group.
3. Recalls about Center-affine invariants In this section we are interested in center-affine invariants and covariants of A(n, m). As a Gl(n)-module, A(n, m) is a direct sum of the subspaces Sk1 , k ∈ {0, 1, 2, . . . , m}. The algebra K(n, m) is multigraded : % K(n, m) = K(n0 , n1 , . . . , nm , p) n0 ,n1 ,n2 ,...nm ,p∈N
where K(n0 , n1 , . . . , nm , p) is the subalgebra of multi-homogeneous covariants of multi-degree (n0 , n1 , . . . , nm , p), i.e. homogeneous of degree nk with respect to coordinates of (aji1 ,i2 ,...ik ) and of degree p with respect to coordinates of x ∈ V . In the same way, we decompose I(n, m) as follows % I(n, m) = I(n0 , n1 , . . . , nm ). n0 ,n1 ,n2 ,...nm ∈N
Before giving the fundamental theorem of center-affine invariants, let us recall the definitions of two fundamental operations over tensors which are the contraction
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and the alternation. In the following definitions, we identify each element of a vector space with its coordinates in a selected basis. Definition 3. A contraction over the tensorial space V ⊗p ⊗ V ∗⊗q is the linear map ϕ : V ⊗p ⊗ V ∗⊗q &−→ V ⊗(p−1) ⊗ V ∗⊗(q−1) defined by j ...%j ...j
ϕ(v)i11...%ikl ...ipq =
n !
j ...j
hj
...j
p l−1 l+1 vi11...ik−1 hik+1 ...iq .
h=1
If p = q, a sequence of contractions ϕ1
ϕp
ϕ2
ϕp+1
V ⊗p ⊗ V ∗⊗p &−→ V ⊗(p−1) ⊗ V ∗⊗(p−1) &−→ · · · &−→ V ⊗ V ∗ &−→ C[V ⊗p ⊗ V ∗⊗q ] is called a complete contraction. For each pair of natural numbers (l, k) with 1 ≤ l ≤ p, 1 ≤ l& ≤ q we get one contraction. Definition 4. A contravariant (respectively covariant) n-vector is an element of V ⊗n (respectively V ∗⊗n ) whose coordinates (εi1 i2 ···in ) (εi1 i2 ···in ) are defined by 1 if (i1 i2 · · · in ) is an even permutation of (12 · · · n) −1 if (i1 i2 · · · in ) is an odd permutation of (12 · · · n) εi1 i2 ···in = εi1 i2 ···in = 0 otherwise
Definition 5. A contravariant (covariant ) alternation over the tensorial space V ⊗p ⊗ V ∗⊗q is the map ϕ : V ⊗p ⊗ V ∗⊗q &−→ V ⊗(p) ⊗ V ∗⊗(q−n) that is defined by
(ϕ : V ⊗p ⊗ V ∗⊗q &−→ V ⊗(p−n) ⊗ V ∗⊗(q) )
j ...j ϕ(v)i11...%h1p...%h2 ......%hn ...iq
=
n !
h1 =1
*
j ...%h ...%h2 ......%hn ...jp
ϕ(v)i11...iq1
=
n !
h1 =1
··· ···
n !
j ......j
p vi11...h1 ...h εh1 ···hn 2 ......hn ...iq
hn =1 n !
hn =1
j ...h ...h2 ......hn ...jp
vi11......i1 q
εh1 ···hn
+
Computation of Affine Covariants of Differential Systems
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Now, we are able to give the fundamental theorem of the classical theory of invariants : Theorem 1. ([3], p.188-189) The expressions obtained with the help of succesive alternations and complete contraction over the tensorial products 1 ⊗nm (S01 )⊗n0 ⊗ (S11 )⊗n1 ⊗ · · · ⊗ (Sm ) ⊗ V ⊗r ⊂ V ∗⊗p ⊗ V ⊗q
with p = (n0 + n1 + · · · + nm + r) and q = (n1 + 2n2 + · · · + mnm ) form a system of generators of K(n0 , n1 , . . . , nm , r). Such polynomials are called basic covariants. Example : For instance, if n = m = 2, the polynomials aαp aβαq aγβγ εpq and aαγ aβδ aγαβ xδ belong respectively to I(0, 1, 2, 0) and K(0, 2, 1, 1). We shall come back to this case in the third section. In order to have a complete contraction, the exponents (n0 , n2 , . . . , nm , r) have to verify the relation n0 + r − (n2 + 2n3 + · · · (m − 1)nm ) = sn with s integer . The above theorem is known as the fundamental theorem of the classical theory of invariants [3]. It can be presented with the contractions and determinants (alternations) ([13, 2]) of contravariant and covariant vectors obtained from the tensors of Sk1 and V by symbolic decomposition. It defines a process of construction of covariants of A(n, m), degree by degree and gives for each suitable choice of alternations and contractions an element of C[A(n, m) × V ]. To get a minimal system of generators we need to know an upper bound of degrees of these generators which we denote by β(n, m). This bound has been calculated by V. Popov [8] and recently improved by H. Derksen [1]. From our point of view, it is still too large for concrete examples. Indeed, let σ(2, 2) be the smallest integer d with the following property : if a ∈ A(2, 2) and 0 doesn’t lie in the Zariski closure of the GL(2, C)-orbit of a, then there exists a non-constant homogeneous invariant I of degree ≤ d such that I(a) .= 0. In the case of planar quadratic differential systems (the list of a minimal system of generators of I(2, 2) is given in the section 5 and the system of generators of the ideal of sygyzies is given in [10]), σ(2, 2) = 6 and dim(I(2, 2)) = 9. Following the first author, β(2, 2) ≤ 12.LCM (1, 2, . . . , σ(2, 2)) = 12.60 = 720 where LCM is the least common multiple. Following the second author, β(2, 2) ≤ max(σ(2, 2),
3 243 dim(I(2, 2))σ(2, 2)2 ) = . 8 2
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We should not forget that the dimension of the linear space A(2, 2) is 12 and so, computations over polynomials depending on 12 indeterminates become fastly complicated. Truly, β(2, 2) = 7. Let us return to computation of center-affine covariants. Actually, it is sufficient to consider the basic covariants with either the contravariant n-vectors or the covariant n-vectors. Indeed, by following relations between the coordinates of n-vectors and those of Kronecker’s symbols , i1 i1 , , δj δj · · · δji1 , n 1 2 , , , , i2 i2 , δj1 δj2 · · · δji2n , , , (3.6) εi1 i2 ···in εj1 j2 ···jn = , . ,, . . . , .. .. . . .. , , , , , i , δ n δ in · · · δ in , j1
j2
jn
any basic covariant can be reduced to sum of basic covariants that contain n-vectors of one kind. Examples. Suppose m = 1 and n = 2. Systems (2.2) become dx = a + Ax. dt where a is an element of V and A, an 2 × 2 matrix.
1. The polynomial Apr Aqs εpq εrs (the half of the determinant of the matrix A) is nothing else that App Ars − Apr Arp = (T race(A))2 − T race(A2 ).
2. The center-affine invariants of (3) containing the vector a and the matrix A are necessairly the multiples of det(a, Aa). Remark 1. Taking into account the Einstein symbolic notation, we remark that a total contraction over V ⊗p ⊗ V ∗⊗p is a polynomial function ϕ : V ⊗p ⊗ V ∗⊗p &−→ C j1 ...jp with ϕ(v) = vjσ(1) ...jσ(p) where σ is a permutation of (1, 2, . . . , p). It follows from this remark that any basic center - affine covariant belonging to K(n0 , n1 , . . . , nm , p) can be represented by the form i
i
il
+1
ai1 · · · ain0 ajn1 0 +1 · · · ajnn01+n1 · · · · · · ajkm−1 +1 ···jk m−1
m−1 +m
il
+n
m · · · ajkm−1 m −m ···jkm
×xilm +1 . . . xilm +p εilm +p+1 ···ilm +p+n · · · εilm +p+(n−1)q+1 ···ilm +p+nq
(3.7)
Computation of Affine Covariants of Differential Systems
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or by the form i
il
i
+1
ai1 · · · ain0 ajn1 0 +1 · · · ajnn01+n1 · · · · · · ajkm−1 +1 ···jk m−1
m−1
il
+n
m · · · ajkm−1 +m m −m ···jkm
×xilm +1 . . . xilm +p εjkm +1 ···jkm +n · · · εjkm +p+(n−1)q" +1 ···jkm +nq" .
(3.8)
where (j1 , . . . , jkm ) and (j1 ,. . . , jkm +nq" ) represent the all possible permutations of -ı s (i1 , . . . , ilm +p+nq ) and ls = r=0 nr , ks = r=1 nr . Of course, we have lm + p + nq = km + nq & . The writtings (3.7) and (3.8) are used to expand a center-affine basic covariant.
4. Affine covariants In this section we show that the knowledge of a system of generators of the algebra of center-affine invariants I(n, m) induces that one of a system of generators the algebra of affine invariants. Actually, this correspondance is an isomorphism. To construct it we need the following lemma. Lemma 1. The group T (n) defines a polynomial representation on the vector space A(n, m). Proof. It suffices to prove that
∀p, q ∈ T (n), ρ2 (p + q) = ρ2 (p)ρ2 (q).
(4.9)
By formula 2.4, we have [ρ2 (p + q)(a)]jα1 α2 ...αk = m−k r # !!
k+r r
r=0 s=0 [ρ2 (p)ρ2 (q)(a)]jα1 α2 ...αk m−k !# i=0
k+i i
$# $ r ajα1 α2 ...αk β1 β2 ...βs γ1 γ2 ...γr−s pβ1 . . . pβs q1γ . . . q γr−s , s
=
$ m−k−i ! # l=0
k+i+l l
$
ajα1 α2 ...αk β1 β2 ...βi γ1 γ2 ...γl pβ1 . . . pβi q γ1 . . . q γl .
We remark that the coefficients of algebraic form ajα1 α2 ...αk β1 β2 ...βi γ1 γ2 ...γl pβ1 . . . pβi q γ1 . . . q γl (k + i + l)! . Hence, the equality (4.9). k!i! Recall that the actions of the groups Gl(n) and T (n) over V are respectively defined by (P, x) &−→ P −1 x and (p, x) &−→ x − p. in both expressions are equal to
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Theorem 2. A polynomial Q ∈ C[A(n, m) × V ] is an Aff (n)-covariant if and only if it exists a Gl(n)-invariant I ∈ I(n, m) such that Q(a, x) = I(ρ2 (x)(a)), ∀ (a, x) ∈ A(n, m) × V. Proof. Let us consider a Gl(n) - invariant I of A(n, m) and let Q be the polynomial Q(a, x) = I(ρ2 (x)(a)). We can suppose that I is a basic invariant. Following (3.8), (3.8) and theorem 1 the polynomial Q(a, x) is a sum of basic center-affine covariants. It remains to prove that Q(a, x) is a T (n) - invariant. By lemma (1), Q(ρ2 (p)(a), x − p) = I(ρ2 (x − p)ρ2 (p)(a)) = Q(a, x). Let us suppose now that Q(a, x) is an Aff (n)-covariant and put I(a) = Q(a, 0). Then I(a) is a Gl(n)-invariant. So, substituting p = x in the relation Q(ρ2 (p)(a), x − p) = Q(a, x), ∀p ∈ T (n), ∀(a, x) ∈ A(n, m) × C n , we obtain I(ρ2 (x)(a)) = Q(ρ2 (x)(a), 0) = Q(a, x).∀ x ∈ T (n), ∀ a ∈ A(n, m) × C n , Thus, the correspondance I(a) ←→ I(ρ2 (x)(a)) = Q(a, x) gives an isomorphism which we denote Φ, between the C - algebras I(n, m) and Q(n, m). The affine covariant Φ(I), where I ∈ I(n, m), denoted I(x) is called sometimes the “translated” of I. In the next section, we compute effectively a minimal system of generators of Q(2, 2). Definition 6. A basic affine covariant is an image by Φ of a basic center-affine invariant. Example : the polynomial aαα
+
2aααβ xβ
+ ··· +
#
$ m+1 α aαβ1 ···βm−1 xβ1 · · · xβm−1 1
is a basic affine covariant. It corresponds to the divergence of the vector field defined by the system. Remark 2. Let I be a center-affine invariant of A(n, m). If it is multi-homogeneous of multi-degree (d0 , d1 , . . . , dm ) then the affine covariant Q(a, x) = I(x.a) has the expansion Q(a, x) = I(a) + Q1 (a, x) + · · · + Qs (a, x) where the polynomial Qi (a, x) is homogeneous of degree i with respect to coordinates m ! x and s = (m − j)di . j=0
Computation of Affine Covariants of Differential Systems
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Corollary 1. A family F of Gl(n)-invariants is algebraically dependant if and only if Φ(F) is algebraically dependant. It is clear that the previous theorem remains true if we substitute the group GL(n) by any of its subgroups like the orthogonal one, O(n). In the particular case where this subgroup contains only identity, we have : Corollary 2. The family of polynomials . / [ρ2 (x)(a)]jα1 α2 ...αk , 1 ≤ α1 ≤ . . . ≤ αk ≤ n, k ∈ {0, . . . , m}, j ∈ {0, . . . , n}
forms a minimal polynomial system of generators of T (n)-covariants of A(n, m).
5. The case of planar quadratic differentiel systems Many works are devoted to the study of center-affine invariants of systems dxj = aj + ajα xα + ajαβ xα xβ (j, α, β = 1, 2) dt
(5.10)
They are based on the classical approach, called Aronhold’s symbolic method ([3]) and resumed in [10]. In particular, a minimal system of generators of I(2, 2) was obtained : Minimal system of generators of I(2, 2)
J1 = aαα , J2 = aαβ aβα , J3 = aαp aβαq aγβγ εpq , J4 = aαp aββq aγαγ εpq , J5 = aαp aβγq aγαβ εpq , J6 = aαp aβγ aγαq aδβδ εpq , J7 = aαpr aβαq aγβs aδγδ εpq εrs , J8 = aαpr aβαq aγδs aδβγ εpq εrs , J9 = aαpr aββq aγγs aδαδ εpq εrs , J10 = aαp aβδ aγµ aδαq aµβγ εpq , J11 = aαp aβqr aγβs aδαγ aµδµ εpq εrs , J12 = aαp aβqr aγβs aδαδ aµγµ εpq εrs , J13 = aαp aβqr aγγs aδαβ aµδµ εpq εrs , J14 = aαp aβr aγαq aδβs aµγδ aνµν εpq εrs , J15 = aαpr aβqk aγαs aδδl aµβγ aνµν εpq εrs εkl , J16 = aαp aβr aγδ aδαq aµβs aνγτ aτµν εpq εrs , J17 = aααβ aβ ,
J18 = apα aα aq εpq ,
J19 = aαβ aβαγ aγ ,
J20 = aαγ aβαβ aγ ,
J21 = apαβ aα aβ aq εpq , J22 = aααβ aβγδ aγ aδ , J23 = aαβγ aβαδ aγ aδ , J24 = aαγ aβδ aγαβ aδ , J25 = aααp aβγq aγβδ aδ εpq , J26 = aααp aβδq aγβγ aδ εpq ,
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D. Boularas
J27 = apα aαβγ aβ aγ aq εpq , J28 = aαβ aβαγ aγδµ aδ aµ , J29 = aαγ aβαβ aγδµ aδ aµ , J30 = aαp aβαq aγβδ aδγµ aµ εpq ,
J31 = aαp aβαq aγβµ aδγδ aµ εpq ,
J32 = aαp aββq aγαµ aδγδ aµ εpq , J34 = aαµp aβαq aγβν aδγδ aµ aν εpq ,
J33 = aαβν aβαγ aγδµ aδ aµ aν , J35 = aαp aβν aγαq aδβµ aµγδ aν εpq ,
J36 = aαpr aβνq aγαs aδβγ aµδµ aν εpq εrs . With the help of tranvectants it was established by N. Vulpe [11] that any element of I(2, 2) is a polynomial of J1 , . . . , J36 . In [9], it was given a minimal system of generators of the ideal of the syzygies of I(2, 2). Minimal system of generators of K(2, 2) Now, let us show that we can deduce from this family of center-affine invariants a minimal system of generators of K(2, 2). Obviously, the proposed method can be applied for any degree m. It is clear that if we substitute aj by xj in the invariants J17 , J18 , . . . , J36 , we obtain
center-affine covariants that we note respectiveley K1 , K2 , . . . , K20 . Actually, we can generalize this procedure of substitution to get all other covariants. Introduce the new center-affine covariant
K21 = det(aj , xj ) = a1 x2 − a2 x1 . Consider a basic center-affine invariant I(ai , aij , . . .) whose notation is (3.7) or (3.8). This is a function of ai and other coordinates ajα1 α2 ,...,αk . There are different ways to replace p times the vector (aj ) by (xj ). For example, if I = apαβ aq aα aβ εpq and p = 1, the substitution may give apαβ xq aα aβ εpq or apαβ aq aα xβ εpq . We shall show that, in general, these two covariants are the same modulo K21 . Let us denote K & (ai , xi , . . .) and K && (ai , xi , . . .) two basic center-affine homogeneous covariants of multi- degree (n0 −p, n1 , . . . , nm , p) obtained from the tensorial notation (3.7) (or (3.8) of I by replacing “p” a = (aj ) by “p” x = (xj ). Then K & (ai , ai , . . .) − K && (ai , ai , . . .) = I(ai , . . .) − I(ai , . . .) = 0. Because of multi-homogeneity with respect to coordinates aj and xj , we deduce from this fact that the difference K & (ai , xi , . . .) − K && (ai , xi , . . .) contains as a factor the covariant a1 x2 − a2 x1 = K21 . Consequently, we have the following result
Computation of Affine Covariants of Differential Systems
13
Lemma 2. Let I be a multi-homogeneous basic center-affine invariant of multidegree (n0 , n1 , . . . , nm ) and p some natural number such that p ≤ n0 . For any two center-affine covariants K & , K && obtained from the invariant I by replacing in its tensorial notation “p” vectors (aj ) by “p” vectors x = (xj ), the difference K & − K && is a multiple of K21 . That is to say that each center-invariant J1 , . . . , J36 of multi-degree (n0 , n1 , n2 , 0) gives rise, for all p ∈ {1, . . . , n1 − 1} to one and only one covariant of multi-degree (n0 − p, n1 , n2 , p) that cannot be expressed as a polynomial of covariants of lower total degree. Applying this lemma to the family {Ji , i = 1 · · · 36} we obtain a polynomialy independant familly of covariants K(2, 2) : K1 = aααβ xβ , K2 = apα xα xq εpq , K3 = aαβ aβαγ xγ , K4 = aαγ aβαβ xγ , K5 = apαβ xα xβ xq εpq , K6 = aααβ aβγδ xγ xδ , K7 = aαβγ aβαδ xγ xδ , K8 = aαγ aβδ aγαβ xδ , K9 = aααp aβγq aγβδ xδ εpq , K10 = aααp aβδq aγβγ xδ εpq , K11 = apα aαβγ xβ xγ xq εpq , K12 = aαβ aβαγ aγδµ xδ xµ , K13 = aαγ aβαβ aγδµ xδ xµ , K14 = aαp aβαq aγβδ aδγµ aµ εpq , K16 = aαp aββq aγαµ aδγδ aµ εpq , K18 = aαµp aβαq aγβν aδγδ xµ xν εpq ,
K15 = aαp aβαq aγβµ aδγδ xµ εpq , K17 = aαβν aβαγ aγδµ xδ xµ xν , K19 = aαp aβν aγαq aδβµ aµγδ xν εpq ,
K20 = aαpr aβνq aγαs aδβγ aµδµ xν εpq εrs , K21 = ap xq εpq ,
K22 = apα aα xq εpq ,
K23 = aα aβ apαβ xq εpq , K24 = aα aβαβ aγγδ xδ , K25 = aα aβαγ aγβδ xδ , K26 = aα aβ apγ aγαβ xq εpq , K27 = aα aβγ aγβδ aδαµ xµ ≤ α2 , K28 = aα aβδ aγβγ aδαµ xµ , K29 = aα aβ aγδν aδγµ aµαβ xν , K30 = aα aβαp aγβq aδγν aµδµ xν εpq ,
K31 = ap aqαβ xα xβ εpq ,
K32 = ap aqα aαβγ xβ xγ εpq , K33 = aα aβαγ aγβδ aδµν xµ xν , Theorem 3. The set of covariants {Ji ; i = 1, . . . , 36}
0
{Kj ; j = 1, . . . , 33}
forms a minimal system of generators of the ring K(2, 2). Proof. We have seen that this family is algebraically independant (consequence of the previous lemma and the procedure of construction of the covariants {Ki ; i =
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D. Boularas
1, . . . , 33}). It is easy to show that it generates the algebra K(2, 2). Let K be a center-affine covariant of multi-degre (n0 , n1 , n2 , p). Following theorem 1, we can suppose that K = K(xi , ai , . . .) is a basic covariant i.e. it is of the form (3.7) or (3.8). As the family {J1 , J2 , . . . , J36 } generates the algebra of center-affine invariants I(2, 2), the invariant I = K(ai , ai , . . .) is a polynomal expression of these invariants : I = P(J1 , J2 , . . . , J36 ) = λ1 I1 + λ2 I2 + . . . λl Il where λi are constant coefficients and Ij are basic invariants of multi-degree (n0 + p, n1 , n2 , 0). Each term of I is a product of invariants Ji . After substituting p (ai ) by p (xi ), we obtain a center-affine covariant of multi-degree (n0 , n1 , n2 , p). Of course, we can choose this substitution in such a way that the obtained covariant is a product of K1 , K2 , . . . , K33 . Let us denote this covariant K & It has the same multidegree that K. The difference ˜ be the quotient of K − K & by K21 . Its multiK − K & is a multiple of K21 . Let K degree is (n0 − 1, n1 , n2 , p − 1). We repeat this procedure . . . At the end, we obtain a polynomial expression of K with respect to J1 , J2 , . . . , J36 , K1 , K2 , . . . , K33 . Remark 3. In the literrature, we often find quadratic systems without free terms : dxj = ajα xα + ajαβ xα xβ (j, α, β = 1, 2). dt The family {Ji ; i = 1, . . . , 16}
0
(5.11)
{Kj ; j = 1, . . . , 20}
is a minimal system of covariants of these systems. This explains the indexation of covariants suggested in the previous theorem. Remark 4. Concerning the affine invariants, it appears that they are nothing more than the polynomial expressions of affine covariants which do not depend over the vector x. They form a subalgebra of Q(n, m). Examples
= aαα + xα aβαβ = aαp aβαq aγβγ *pq + 2aαpδ aββq aγβγ xδ *pq = aαp aβγq aγαβ *pq + 2aαpδ aβγq aγαβ xδ *pq ··· α β γ δ J13 (x) = ap aqr aγs aαβ aµδµ *pq *rs + 2aαpu aβqr aγγs aδαβ aµδµ xu *pq *rs J1 (x) J3 (x) J5 (x)
= J1 + 2K1 , = J3 + 2(K9 − K10 ), = J5 + 2(K9 − K10 ),
(5.12)
= J13 − 2J7 K1 .
Taking into account the relation J7 (x) = J7 , it is obvious that the polynomials J3 − J5 and J1 J7 + J13 are affine invariants.
Computation of Affine Covariants of Differential Systems
15
6. Effective computation of a system of generators of Q(2, 2)
In this section we deal with the differential systems (5.10). The goal is to compute effectively a system of generators of Q(2, 2) from that one of I(2, 2) (see the previous section). The algorithms are implanted in Maple 5.4. Some of algorithms presented here can be generalised to any dimension of differential systems. 6.1. Computing in K(2, 2). The algebra of center-affine covariants of quadratic differential systems (5.10), K(2, 2) is a multigraded algebra : % % K(2, 2) = K(n0 , n1 , n2 , p) l∈N n0 +n1 +n2 +p=l
where K(n0 , n1 , n2 , p) is the subalgebra of multi-homogeneous covariants of multidegree (n0 , n1 , n2 , p), i.e. homogeneous of degree nk with respect to cordinates of (aji1 ,i2 ,...ik ) and of degree p with respect to coordinates xj . As it is proved previously, the family of covariants F = {J1 , . . . , J36 , K1 , . . . , K33 } is a minimal system of generators of K(2, 2). The main task which arises here is to express any center-affine covariant (invariant) in this basis. For this, we need to introduce the notion of type of basic center-affine covariant : this is the sextuple (p, q, r, s, τ, v) where (p, q, r, s) is its multi-degree with repect to coordinates (ai , aij , aij,k , xi ), τ is the number of contravariant 2-vectors εjk and, v, following the remark (1), is the permutation of upper indices of the covariant. Algorithm 1.1 : expansion of a basic covariant of type (p, q, r, s, τ, v). We take into account three facts : 1. the coordinates (aijk ) are symmetric with respect to subscripts, 2. if n = 2, εij = εij = j − i 3. the expansion of a basic covariant is obtained by successive contractions. The procedure is called devcov. Algorithm 1.2 : decomposition of the multi-degree m w.r.t. the list of multidegrees of F. This list, denoted t, has 49 elements : t :=
[[0, 1, 0, 0], [1, 0, 1, 0], [0, 2, 0, 0], [2, 1, 0, 0], [1, 1, 1, 0], [0, 1, 2, 0],
16
D. Boularas
[3, [3, [2, [1, [0, [1, [1, [0,
0, 1, 0, 0, 2, 0, 1, 0,
1, 1, 4, 0, 1, 4, 1, 3,
0], [2, 0], [2, 0], [1, 1], [0, 1], [0, 1], [0, 2], [0, 3]].
0, 1, 2, 0, 0, 2, 1,
2, 2, 3, 1, 3, 3, 2,
0], [1, 2, 1, 0], [1, 0, 3, 0], [0, 2, 2, 0], [0, 0, 4, 0], 0], [1, 3, 1, 0], [0, 3, 2, 0], [0, 1, 4, 0], [3, 0, 3, 0], 0], [1, 0, 5, 0], [0, 2, 4, 0], [0, 0, 6, 0], [0, 3, 4, 0], 1], [1, 1, 0, 1], [0, 1, 1, 1], [2, 0, 1, 1], [1, 0, 2, 1], 1], [2, 1, 1, 1], [1, 1, 2, 1], [0, 1, 3, 1], [2, 0, 3, 1], 1], [0, 0, 5, 1], [0, 1, 0, 2], [1, 0, 1, 2], [0, 0, 2, 2], 2], [1, 0, 3, 2], [0, 0, 4, 2], [0, 0, 1, 3], [0, 1, 1, 3],
The decomposition of the multi-degree m following the list t is given by the procedure reduire which is presented in the annex 2. Let [p, q, r, s] the multi-degree of center-affine covariant. This procedure gives all possible decompositions [p, q, r, s] = -49 1 λi t[i].
Algoritm 1.3 : construction of a generating family of K(p, q, r, s) (as a linear space), starting from F. With the help of the previous algorithm, we get the different decompositions of the 4-tuple K(p, q, r, s) w.r.t. the family F. As we know that the dimension of spaces K(t[1]), K(t[2]), K(t[3]), K(t[4]), K(t[7]), K(t[9]), K(t[11]), K(t[13]), K(t[16]), K(t[18]), K(t[19]), K(t[20]), K(t[21]), K(t[22]), K(t[23]), K(t[24]), K(t[25]), K(t[26]), K(t[27]), K(t[29]), K(t[31]), K(t[33]), K & (t[36]), K & (t[37]) is one, that one of K(t[5]), K(t[8]), K(t[10]), K(t[14]), K(t[28]), K(t[30]), K(t[32]), K(t[34]), K(t[42]), K(t[44]) is two and dimension of spaces K(t[6]), K(t[12]), K(t[15]), -49 K(t[17]), K(t[35]) is three, for each term of the decomposition [p, q, r, s] = 1 λi t[i] we take the linear basis of K(t[i]) and so, we form the generating family of the linear space K(p, q, r, s) (w.r.t. to the family F). For example, for K(0, 2, 4, 0), the algorithm 1.3 gives us w.r.t. t, the decompositions [0, 2, 4, 0]
= 2[0, 1, 0, 0] + [0, 0, 4, 0] = [0, 1, 0, 0] + [0, 1, 4, 0] = [0, 2, 0, 0] + [0, 0, 4, 0] = 2[0, 1, 2, 0] = [0, 2, 2, 0].
For the first decomposition, we have J[1]2 J[7], J[1]2 J[8], J[1]2 J[9], for the second, J[1]J[11], J[1]J[12], J[1]J[13], for the third, J[2]J[7], J[2]J[8], J[2]J[9], for the fourth, J[3]2 , J[3]J[4], J[3]J[5], J[4]2 , J[4]J[5], J[5]2 and for the last, J[14]. These 16 covariants (here, invariants) form a generating family of K(0, 2, 4, 0). This algorithm corresponds to the procedure dcpcov. Algoritm 1.4 : expression of any covariant in the basis F. This task is reduced to solving of a linear system. Before to do it, we can operate main simplifications. According to [1, 12] any center-affine covariant K of any
Computation of Affine Covariants of Differential Systems
17
tensorial representation of the group GL(2, C) can be written as an homogeneous polynomial in C[A(2, 2)][x1 , x2 ] of degree p : K = A11···11 (x1 )p + A11···12 (x1 )p−1 x2 + · · · + A22···22 (x2 )p . With the help of some operator, the author shows that K, as a polynomial in [x1 , x2 ], vanishes if and only if the leading coefficient A11···11 (called elsewhere the semi-invariant), vanishes. That means that the leading coefficient define uniquely the whole covariant. This fact is also true for any homogeneous relation between covariants. This algorithm corresponds to the procedure ExpCovAff. Note that this procedure gives also the syszygies of multidegree (p, q, r, s). 6.2. Computing of a minimal system of generators of Q(2, 2). The goal of this subsection is to compute a minimal system of generators of Q(2, 2) using the center-affine covariants I(2, 2) which is given in the previous subsection. We have noted it F. Following the theorem (2), the image G = φ(F) is a minimal system of generators of Q(2, 2). To have their expanded expressions, it suffices to make the substitutions ai ← ai + aiα xα + aiαβ xα xβ , aij ← aij + 2aijα xα , aijk ← aijk in F. The minimal system of generators of Q(2, 2) is then known. It remains to express these covariants in F. This problem is the same that the determination of their multi-homogeneous parts. Let J be a center-affine invariant. We decompose the corresponding affine covariant Φ(J) into multi-homogeneous polynomials Φ(J) = L1 + · · · + Lk . The different steps for computing Φ(J) are the following : 1. determination of the multi-degrees of Φ(J) ( algoritm 2.1) w.r.t. the list of variables varH := [{A1 , A2 }, {A1, 1 , A1, 2 , A2, 1 , A2, 2 }, {A1, 11 , A1, 12 , A2, 11 , A2, 12 , A1, 22 , A2, 22 }, {x1 , x2 }]
18
D. Boularas
2. determination of the multi-homogeneous part Li of Φ(J) corresponding to a giving multi-degree (algoritm 2.2, algoritm 2.3). 3. expression of each multi-homogeneous part Li of Φ(J) w.r.t. F. Algoritm 2.1 : Multi-degrees of affine covariants. Let J be center-affine invariant. Its type is (p, q, r, 0). Each affine covariant of planar quadratic systems is a sum of (p& , q & , r& , s& ) - multi-homogeneous center-affine covariants. For finding these multi-degrees, it suffices (algorithm 2.1) to expand the univariate polynomial pp := (a + b × x + c × x2 )p × (b + 2 × c × x)q × cr and to take the list of elements (degree(pp, a), degree(pp, b), degree(pp, c), degree(pp, x)) . This procedure is called DegHom. Algoritm 2.2 : this algorithm tests whether a monomial P has a multi-degree n = (n1 , n2 , n3 , n4 ) w.r.t. the list of the variables varH. This procedure is called DegMonHom. Algoritm 2.3 : let P be an affine covariant. Following the remark (2), P is a sum of multi-homogeneous center-affine covariants. The procedure given by this algorithm (called PartHom) consists in the determination of its multi-homogeneous parts. Algoritm 2.4 : this algorithm (that uses the previous ones) gives the expression of any basic affine covariant relatively the familyF. The corresponding procedure is called DevCovAff.
Acknowledgments The author wishes to express his sincere thanks to an anonymous referee for carefully reading the manuscript and contributing many useful remarks leading to an improvement of the presentation of this paper. Thanks also to Jacques-Arthur Weil for inspiring discussions on implementation issues.
Computation of Affine Covariants of Differential Systems
19
7. Annexes 7.1. Algorithms (Maple 5.5). All algorithms are implemented in Maple 5.5. They form the package denominated SIB. 7.1.1
Computing in K(2, 2)
Algorithm 1.1 : expansion of a basic covariant of type (p, q, r, s, τ, v). devcov := proc(p, q, r, s, τ, v) local α, j, I1 , I2 , I3 , Xx , Eps1 , Eps2 , J, JJ ; I1 := 1;
I2 := 1;
I3 := 1;
global A, x;
Xx := 1; Eps1 := 1;
Eps2 := 1;
α := vector(p + q + r + s + 2 × τ ) ; for j to p do I1 := I1 × Aαj od;
for j to q do I2 := I2 × Aαp+j , αvj od; for j to r do
I3 := I3 × Aαp+q+j ,10×min(αvq+2×j−1 , αvq+2×j )+max(αvq+2×j−1 , αvq+2×j ) for j to s do Xx := Xx × xαp+q+r+j od ;
for j to τ do Eps1 := Eps1 × (αp+q+r+s+2×j − αp+q+r+s+2×j−1 ) od ; for j to 1/2 × p + 1/2 × s − 1/2 × r + τ do
Eps2 := Eps2 × (αvq+2×r+2×j − αvq+2×r+2×j−1 ) od;
J0 := Eps1 × Eps2 × I1 × I2 × I3 × Xx ; for j to p + q + r + s + 2 × τ do
Jj := subs(αj = 1, Jj−1 ) + subs(αj = 2, Jj−1 ) od ;
JJ := sort(eval(Jp+q+r+s+2×τ ), [A1 , A2 , A1, 1 , A1, 2 , A2, 1 , A2, 2 , A1, 11 , A1, 12 , A1, 22 , A2, 11 , A2, 12 , A2, 22 , x1 , x2 ]); collect(JJ , [x1 , x2 ]) end
od;
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D. Boularas
Algorithm 1.2 : decomposition of a multi-degree following the list t. intquo := proc(i1 , i2 ) if i2 = 0 then RETURN(∞) else RETURN(iquo(i1 , i2 )) fi end reduire1 := proc(m, t, M, e) local
i, k;
global
T;
for k to max(seq(mi , i = 1..nops(m))) do if m − k × t = [0 $ nops(m)] then RETURN(M + k × Te ) else next fi od;
RETURN(NULL) end reduire := proc(m, t, M, e) local
i, j; global
T;
if {op(m)} = {0} then RETURN(M ) fi ;
if nops(t) .= 1 then
i := min(seq(intquo(mj , t1j ), j = 1..nops(m))) ; seq(reduire(m − j × t1 , subsop(1 = NULL, t), M + j × Te , e + 1), j = 0..i)
else RETURN(reduire1(m, t1 , M, e)) fi end
Computation of Affine Covariants of Differential Systems
21
Algorithm 1.3 : it constructs a basis of the linear space K(p, q, r, s), starting from F. dcpcov := proc(m) g, i, j, k, l, H, HH , J, R, S;
local
global
M, T, U, t;
S := {} ; R := subs(M = 0, {reduire(m, t, M, 1)}) ; J := vector(nops(R)) ;
for l to nops(R) do H := {1} ;
Jl := map(op, indets(Rl )) ;
for k to nops(Jl ) do if has({11, 16, 13, 41, 43, 45, 46, 47, 48, 49, 1, 2, 3, 4, 7, 9, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 31, 33, 36, 37, 38, 39, 40}, Jlk ) then H := {seq(Hj × UJlk , 1 coeff(Rl , TJlk ) , j = 1..nops(H))}; fi;
if has({10, 14, 42, 44, 5, 8, 28, 30, 32, 34}, Jlk ) then HH := {} ;
HH := {seq(UJlk , 1 i × UJlk , 2 (coeff(Rl , TJlk )−i) , i = 0..coeff(Rl , TJlk ))}; H := {seq(seq(Hj × HH i , j = 1..nops(H)), i = 1..nops(HH ))} fi;
if has({15, 12, 6, 17, 35}, Jlk ) then HH := {} ;
for i from 0 to coeff(Rl , TJlk ) do for g from 0 to coeff(Rl , TJlk ) do if coeff(Rl , TJlk ) − i − g < 0 then elseHH := HH fi;
od;
od;
next
union {UJlk , 1 × UJlk , 2 g × UJlk , 3 (coeff(Rl , TJlk )−i−g) } i
H := {seq(seq(Hj × HH i , j = 1..nops(H)), i = 1..nops(HH ))} fi;
od;
S := S od;
S;
union end
eval(H, 1)
D. Boularas
22
Algoritm 1.4 : expression of any covariant in the basis F. ExpCovAff := proc(cv , CV ) local
S, Ss, SS , SV , CVV , i, varT , delpol , sys, sosol , ind , α;
global
V, A, x, U, K;
SS := dcpcov(cv ) ;
S := [seq(SS i , i = 1..nops(SS ))] ;
SV := subs(U = V, S) ; SV := subs(x1 = 1, x2 = 0, eval(SV )) ; CVV := subs(x1 = 1, x2 = 0, eval(CV )) ; varT := [A1 , A2 , A1, 1 , A1, 2 , A2, 1 , A2, 2 , A1, 11 , A1, 12 , A1, 22 , A2, 11 , A2, 12 , A2, 22 ]; ind := {seq(αi , i = 1..nops(SV ))} ; delpol :=
collect(add(αi × SV i , i = 1..nops(SV )) − CVV , varT , distributed ) ; sys := {coeffs(delpol , varT )} ;
sosol := solve(sys, ind ) ;
Ss := add(subs(sosol , αi ) × Si , i = 1..nops(S)) ;
RETURN(subs(seq(αj = 0, j = 0..nops(ind )), Ss));
7.1.2
end
Computing of affine covariants
Algorithm 2.1 : list of multi-degree of terms of an affine covariant. DegHom := proc(p, q, r, s) local
l, pp, i, H;
H := {} ; pp := expand((a + b × x + c × x2 )p × (b + 2 × c × x)q × cr ) ; if type(pp, monomial ) = true then H := {[p, q, r, s]}else
l := convert(pp, list) ; for i to nops(l) do
H := H union {[degree(li , a), degree(li , b), degree(li , c), degree(li , x)]} od; H; fi end
Algorithm 2.2 : we test if the monomial P has the multidegree n w.r.t. varH.
Computation of Affine Covariants of Differential Systems
23
DegMonHom := proc(P, varH , n) if varH = [ ] then RETURN(true) fi ; if degree(P, varH 1 ) = n1 then if nops(varH ) = 1 then RETURN(true) else RETURN(DegMonHom(P, subsop(1 = NULL, varH ), subsop(1 = NULL, n))) fi else RETURN(false); fi end
Algorithm 2.3 : determination of the (n1 , n2 , n3 , n4 ) multi-homogeneous part of a given affine covariant P w.r.t. varH. PartHom := proc(P, varH , n) local
PP , L, varTx , i;
varTx := [seq(op(varH i ), i = 1..nops(varH ))] ; PP := collect(P, varTx , distributed ) ; L := select(DegMonHom, convert(PP , list), varH , n) ; convert(L, ‘ + ‘) end
Algorithm 2.4 : expression of a given affine covariant w.r.t. to F.
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D. Boularas
DevCovAff := proc(v, CT ) local
C, VV , Vv , H, vv , i, j, varT ;
C := subs(A = B, CT ) ; VV := collect(subs(B1 = A1 + A1, 1 × x1 + A1, 2 × x2 + A1, 11 × x1 2
+ 2 × A1, 12 × x1 × x2 + A1, 22 × x2 2 , B2 = A2 + A2, 1 × x1 + A2, 2 × x2 + A2, 11 × x1 2 + 2 × A2, 12 × x1 × x2 + A2, 22 × x2 2 ,
B1, 1 = A1, 1 + 2 × A1, 11 × x1 + 2 × A1, 12 × x2 , B1, 2 = A1, 2 + 2 × A1, 12 × x1 + 2 × A1, 22 × x2 ,
B2, 1 = A2, 1 + 2 × A2, 11 × x1 + 2 × A2, 12 × x2 ,
B2, 2 = A2, 2 + 2 × A2, 12 × x1 + 2 × A2, 22 × x2 , B1, 11 = A1, 11 , B1, 12 = A1, 12 , B1, 22 = A1, 22 , B2, 11 = A2, 11 , B2, 12 = A2, 12 , B2, 22 = A2, 22 , C), [x1 , x2 ], distributed ); H := DegHom(v1 , v2 , v3 , v4 ) ; for i to nops(H) do vv i := subs(x1 = 1, x2 = 0, PartHom(VV , varH , Hi )) ; Vv i := ExpCovAff(Hi , vv i ) ; Vv i := subs(seq(αj = 0, j = 0..nops(ind )), Vv i ) ; for j to nops(ind ) do Vv i := subs(αj = 0, Vv i ) od ; Vv i := subs(U1, 1 = J1 , U2, 1 = J17 , U3, 1 = J2 , U4, 1 = J18 , U5, 1 = J19 , U5, 2 = J20 , U6, 1 = J3 , U6, 2 = J4 , U6, 3 = J5 , U7, 1 = J21 , U8, 1 = J22 , U8, 2 = J23 , U9, 1 = J24 , U10, 1 = J25 , U10, 2 = J26 , U11, 1 = J6 , U12, 1 = J7 , U12, 2 = J8 , U12, 3 = J9 , U13, 1 = J27 , U14, 1 = J28 , U14, 2 = J29 , U15, 1 = J30 , U15, 2 = J31 , U15, 3 = J32 , U16, 1 = J10 , U17, 1 = J11 , U17, 2 = J12 , U17, 3 = J13 , U18, 1 = J33 , U19, 1 = J34 , U20, 1 = J35 , U21, 1 = J36 , U22, 1 = J14 , U23, 1 = J15 , U24, 1 = J16 , U25, 1 = K21 , U26, 1 = K1 , U27, 1 = K22 , U28, 1 = K3 , U28, 2 = K4 , U29, 1 = K23 , U30, 1 = K24 , U30, 2 = K25 , U31, 1 = K8 , U32, 1 = K9 , U32, 2 = K10 , U33, 1 = K26 , U34, 1 = K27 , U34, 2 = K28 , U35, 1 = K14 , U35, 2 = K15 , U35, 3 = K16 , U36, 1 = K29 , U37, 1 = K30 , U38, 1 = K19 , U39, 1 = K20 , U40, 1 = K2 , U41, 1 = K31 , U42, 1 = K6 , U42, 2 = K7 , U43, 1 = K32 , U44, 1 = K12 , U44, 2 = K13 , U45, 1 = K33 , U46, 1 = K18 , U47, 1 = K5 , U48, 1 = K11 , U49, 1 = K17 , Vv i ); od; collect(add(Vv i , i = 1..nops(H)), [seq(Ki , i = 0..33)]); end
Computation of Affine Covariants of Differential Systems
7.2. Minimal system of generators of the algebra Q(2, 2). Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15
Q16
Q17 Q18
J 1 + 2 K1 J 2 + 4 K3 + 4 K7 , J3 − 2 K10 + 2 K9 , J4 − 2 K10 , J5 − 2 K10 + 2 K9 , (J5 − 2 J3 + J4 ) K1 + J6 − 2 K16 + 4 K15 − 2 J1 K10 + J1 K9 + 4 K18 , J7 , J8 , J9 , (2 J1 J5 + 8 K18 + 2 J1 K10 − J1 J3 ) K1 + (2 J9 − 6 J8 + 2 J7 ) K2 + (J4 − 2 J3 + 5 J5 − 4 K10 ) K3 + (J4 − 7 J5 ) K4 − 4 J8 K5 + (−8 K9 + 4 K10 − 2 J4 − 4 J5 + 4 J3 ) K6 + (−2 J3 + 8 J5 + 4 K9 ) K7 + (−J12 + 3 J2 ) K9 + (−J12 −J2 ) K10 +J10 +6 K19 −3 J1 K14 +4 J1 K18 +4 J1 K15 −2 J1 K16 , J11 − 2 K20 , (J8 − J7 ) K1 + J12 − 2 K20 , J13 − 2 J7 K1 , (J1 J9 −J1 J8 −2 J12 ) K1 +(−2 J9 −2 J7 ) K3 +2 J8 K4 +(2 J7 +2 J8 ) K6 − 4 J7 K7 + (4 J3 − 2 J5 ) K9 + J14 − 2 J4 K10 , J15 , −4 J7 K13 + (10 J12 − 4 K20 ) K12 + ( 32 J1 J12 − J32 − 16 K9 K10 − 8 J9 K7 + 20 J7 K4 + 8 J8 K6 − 32 J52 + 12 J2 J8 − 8 J8 K3 − 52 J1 J11 + 32 J4 J5 )K1 + (−8 J12 + 4 K20 − 25 J1 J8 + J13 + 7 J11 ) K3 + ( 21 J1 J8 − 5 J11 + 5 J12 − J13 ) K4 −8 J15 K5 +(8 J13 +4 J12 +4 K20 −4 J11 ) K6 +(−16 K20 −J1 J8 + J1 J9 −10 J12 +8 J11 −8 J13 ) K7 +(3 J8 −3 J7 ) K8 +6 J1 K92 +(−24 K16 + 3 2 J J + 12 J1 J3 − J1 J5 − 12 K18 + 8 K14 − 28 J1 K10 ) K9 + 20 J1 K10 + 2 1 4 1 3 3 (20 K16 + 2 J1 J5 −8 K14 +4 K15 − 2 J1 J3 − 2 J1 J4 +24 K18 ) K10 +(6 J8 − 2 J9 ) K12 + (−2 J8 − 10 J7 ) K13 + (3 J3 + 3 J5 − 5 J4 ) K14 + 5 J4 K15 + (J5 − 4 J3 ) K16 + (4 J8 + 16 J9 ) K17 + 4 J5 K18 + (−J2 + 21 J12 ) K20 + J16 , J17 + K4 + K6 , (− 32 K2 − 2 K21 − K5 ) K12 + (−2 J1 K5 − 2 J1 K2 − 4 K22 + K11 + 2 K31 ) K1 + ( 21 J2 + 2 J17 + 2 K3 + 32 K7 − 12 J12 ) K2 + (K5 + 4 K21 ) K3 − 2 K21 K4 + (−J12 + K7 ) K5 + 2 K21 K7 + J1 K11 + (J2 + 2 J17 ) K21 + J18 − 2 K23 − 2 K32 − J1 K22 + 3 J1 K31 , (continued on the next page)
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26 Q19 Q20
Q21
Q22
Q23
Q24
Q25 Q26
Q27
J19 + 2 K25 + K8 − J1 K12 + J1 K7 + 2 K1 K4 − 2 K13 + 3 K12 + 2 K17 , −K13 − 2 J1 K12 + ( 12 J2 − 12 J12 + 2 K3 + K7 + 2 K4 + 2 K6 ) K1 + J20 + 2 K24 + J1 K4 − K13 + 2 J1 K6 , (− 32 J1 K5 − 32 J1 K2 ) K12 + ((− 12 J12 − 3 J17 − 21 J2 − 32 K6 ) K2 + ( 32 K5 − 3 K21 ) K4 + (− 23 J17 − K6 − 2 J12 ) K5 − 32 K21 K6 + 2 J1 K11 + (−3 J17 − 3 J2 ) K21 + 2 J18 + 2 J1 K31 − 2 K32 )K1 + (−3 J1 J17 + 23 J1 K7 − J1 K4 + 3 K12 −3 K13 +K8 +3 J20 +6 K25 +J1 K3 +6 K25 +J1 K3 −2 J1 K6 ) K2 + 3 K22 K3 + (2 K31 + J1 K21 + 2 K11 − 4 K22 ) K4 ( 23 K25 − 3 K13 + 3 K12 + K17 − J1 J2 − J1 K6 ) K5 + (K31 − 2 J1 K21 + K11 ) K6 ( 23 K11 − 3 K22 + 3 J1 K21 ) K7 + 3 K21 K8 + ( 21 J2 + 12 J12 ) K11 + 2 K21 K13 + (2 J1 J17 − 2 J20 + 3 K25 ) K21 − 3 J1 K23 + 3 K26 + (2 J17 + 23 J2 + 32 J12 ) K31 + J21 + 3 K21 K17 − 3 J1 K32 , 2 −J1 K13 + (−J17 − J12 + K3 − 21 K6 ) K12 + (2 J19 + K25 + J1 K3 + J1 K4 + 1 K17 −2 J1 J17 +J1 K6 ) K1 +(−J3 +K10 −2 K9 ) K2 +(2 J17 +K7 ) K4 + 2 (−J4 − 12 K9 ) K5 +K62 +(2 J17 + 12 J2 + 12 J12 ) K6 +J22 −2 K28 +2 J1 K24 − K21 K9 − J1 K13 , −J1 K13 + (− 12 J2 + 2 K4 − 12 K7 ) K12 + (2 K12 + 2 K17 − 2 K13 − J1 J17 + J20 ) K1 + (−J5 + J4 + K10 ) K2 + K3 K7 + K42 + (J17 − K7 ) K4 + (−J5 + K10 ) K5 + 21 K72 + 2 K33 − 2 K28 + 2 K27 + J1 K12 + J23 − J1 K13 − J5 K21 − K6 K7 + J1 K25 + 12 J2 K7 + J1 K17 , −2 K14 + (4 K6 − 54 J2 − K3 ) K12 + (4 K17 − J1 J17 − 3 J1 K7 − J1 K3 + 2 K12 − 21 J1 K4 + 2 J20 + 4 K8 + 4 K24 ) K1 + (−2 K9 + 3 J3 − 21 J5 − K10 ) K2 + K32 + ( 12 J2 + 4 K7 − 2 K4 − 12 J12 ) K3 + 52 K42 + 3 K4 K7 + (4 K10 − J5 ) K5 − 4 K62 + ( 12 J12 − 4 J17 − 12 J2 ) K6 + 2 K72 + (− 14 J12 + 3 J ) K7 + J1 K8 + 4 K21 K9 + 27 J1 K12 − 52 J1 K13 + 3 J1 K17 + (−J4 + 2 2 J5 ) K21 + J24 + 4 K27 + 4 K33 + J1 K25 − 2 K28 , (J3 − 12 J4 + 12 J5 − K10 + 2 K9 ) K1 + J25 − K15 + 12 J1 K9 − K18 , (J3 + K9 ) K1 + J26 − K16 , J27 + (11 K32 + ( 12 J12 − 52 K4 − 12 J2 ) K3 − 2 K42 + ( 72 K7 + 12 J2 − 12 J12 − K6 ) K4 +(−7 J3 +3 J5 ) K5 + 52 K62 +(− 92 J12 +7 J2 ) K6 − 45 K72 +( 72 J12 − 27 J ) K7 + 92 J1 K13 + 6 J1 K17 + (2 J5 − 2 J4 ) K21 − 3 J24 + 2 K28 − 4 2 2 2 K27 − 23 J23 + 92 J17 − 4 J1 K25 )K2 + ((− 25 J12 − 52 K6 + 52 K7 ) K5 + 2 J1 K11 +( 32 J2 − 32 J12 −9 J17 ) K21 −6 K32 ) K4 − 13 K42 K5 +(−7 J1 K11 + 2 (−2 J2 −6 J17 ) K21 + 92 K32 +3 J1 K22 +3 K23 ) K7 +((J12 +K6 − 11 J − 2 2 1 9 1 4 2 K ) K − 4 J K + 4 K + J K ) K + K K − 2 K K 7 5 1 31 32 11 3 5 10 1 5 + 2 2 1 2 (−2 K31 − 12 K11 + 2 K22 + J1 K21 ) K8 + (continued on the next page)
Computation of Affine Covariants of Differential Systems
(12 K24 + 21 J13 − 6 J20 + 4 K25 + 6 J1 J17 − 4 K12 − 5 K13 − 1 2 J J + K17 ) K11 − 3 K21 K10 + 4 K25 K31 − 12 K24 K31 + (−10 K31 + 2 1 2 6 J1 K21 ) K12 + (3 K31 − 27 J1 K2 − 3 J1 K5 ) K13 + (−2 J1 K21 + 20 K31 ) K13 + ((− 25 K6 + 45 K7 + 54 J2 + 72 K3 ) K2 + (3 K5 + K21 ) K3 − 5 J1 K22 − 6 K21 K6 + 27 K32 − 2 J2 K21 − 32 J1 K11 − 3 K6 K5 + 2 K21 K4 )K12 + (−3 K22 − J1 K21 − K31 ) K17 − 6 J4 K21 + 3 K21 K62 + 2 (−K25 + 6 K24 − 2 J19 ) K22 + (J1 K25 + 3 K28 + J17 + 2 J1 J19 − 3 K27 − J23 ) K21 + (− 23 J12 + 32 J2 ) K23 + (3 J17 + 32 J2 − 32 J12 ) K32 + ( 32 J13 − Q27 J + 6 J20 − 32 J1 J2 ) K31 + ((−K24 − 12 J13 + 12 J1 J2 − K13 + 92 K8 − (end) 19 9 J1 K3 + 2 J20 ) K2 + (6 K11 + 6 K22 ) K3 − 4 K22 K4 + (−K8 + 23 J1 K6 + 12 J19 + 12 J1 J2 + 12 J1 K7 − 8 J1 J17 + 3 K25 − 52 K13 ) K5 + 3 J1 K21 K6 − J1 K21 K7 +6 K21 K8 − 21 J2 K11 −21 K21 K13 +6 J19 K21 +( 23 J2 −6 J17 − 3 2 J ) K22 + 2 J21 + 3 J17 K31 )K1 + ( 13 J + 72 K10 − 17 J ) K22 + (2 K62 + 2 1 2 5 2 3 1 9 19 2 2 2 (5 J2 − 5 J1 + K7 ) K6 − 2 K7 + (− 4 J2 + 4 J1 ) K7 + 8 J1 K13 + 41 J22 + 2 J1 K24 − 5 J1 K12 + 12 K28 − 4 J24 − J1 K17 − 6 J12 J17 − 3 K21 K9 − 12 K27 − 14 J14 − 3 K33 + 11 J K8 )K5 + (3 K21 K7 + 6 J1 K11 + (15 J17 + 2 1 6 J2 ) K21 + 3 K32 − 12 K23 + 2 J1 K31 ) K6 , J28 +2 K29 +J1 K27 +2 J1 K33 −J2 K13 −4 K21 K14 +2 J5 K31 +(4 J1 K6 + J1 K4 +J1 K7 −2 K12 +6 K25 ) K3 −J1 K28 +(2 J1 K10 −2 J26 −2 K14 − 4 K15 − 21 J1 J5 − J6 + 2 K16 + 21 J1 J4 ) K2 − 2 J5 K11 + (−2 J1 K6 + J19 + 5 K17 + 21 J1 J17 ) K4 + (−J1 K7 − 14 J13 − 3 J1 K3 ) K12 + (−2 J6 + K14 − 2 J1 J4 ) K5 + (− 21 J1 J2 + 12 J13 − 4 K8 + K12 + 2 K17 − 2 J1 J17 ) K6 + Q28 (2 J20 + K17 + K8 + 2 K25 − 3 J19 + 2 J1 J17 + 21 J1 J2 − 14 J13 + 52 K12 − − 12 J12 ) K25 +(2 J17 + 12 J2 ) K12 +(J3 −J5 ) K22 +((3 J4 −2 K9 +J3 ) K2 + 4 K32 +(K4 − 21 K6 −J17 ) K3 +2 K42 +(−K9 − 21 J3 + 12 J4 ) K5 −K6 K7 + (−J2 − 2 J17 ) K7 − 2 K21 K9 + 2 J1 K12 + (−J3 + J4 ) K21 − 21 J1 J20 − 4 J1 K25 − 12 J12 J17 + 2 J1 K24 + J24 )K1 + (4 J17 + 3 J2 − J12 ) K17 − 1 2 J K13 + ( 12 J1 J5 − J6 ) K21 + J1 K72 , 2 1 −K13 K6 + (− 32 K13 − J1 J17 − 2 K24 − 7 K8 ) K12 + ((−6 J5 + 5 J3 + J4 ) K2 + 3 K32 + 2 K42 + (−J17 − K6 − J12 ) K4 − K10 K5 + K62 + (−3 J12 + 2 K7 ) K6 − J17 − J23 + 3 J12 K7 − 2 K27 − 2 J1 J20 + 2 J22 + J1 K25 + 2 K28 )K1 + (−J6 + J1 J4 − 17 K15 + J25 + 5 K16 + 12 K14 − 4 J26 ) K2 + (9 J1 K6 −12 K12 +K17 +J1 K4 +11 K13 −6 J1 K7 ) K3 +J1 K42 +(K25 + Q29 2 J20 − 2 K13 + 2 J19 − 5 J1 K7 + 2 J1 K6 + 4 K24 ) K4 + (4 J25 − K14 − 8 J26 − K18 − 6 J1 J3 + 3 J1 J4 − 4 K15 − 2 J6 ) K5 + (5 K13 + 4 K24 − 1 J J + 12 J13 − 3 K8 + K17 + 2 K25 ) K6 + (−K12 + 12 K8 + 25 K13 ) K7 + 2 1 2 (K22 + 4 K31 ) K9 + (−4 K11 − 4 K31 ) K10 + 4 J17 K12 + ( 21 J2 − 12 J12 − 2 J17 ) K13 + 2 K21 K18 + 2 J4 K22 + (−J2 + J12 ) K24 + 2 J17 K25 + J29 , (continued on the next page)
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28
Q30
Q31
Q32
Q33
(3 K9 −2 K10 − 12 J4 ) K12 +(−J1 K10 + 21 J1 K9 −J25 +J26 −2 K18 ) K1 + (− 23 J8 − 12 J7 ) K2 +(−2 K9 +2 K10 ) K3 +K4 K9 +(−2 J7 +J9 −J8 ) K5 + (J4 + J5 − J3 ) K6 + (−K9 + 21 J5 ) K7 + J1 K18 + J17 K9 + K19 + 2 K30 − J8 K21 − J17 K10 , (−2 K10 − 12 J3 + 3 K9 ) K12 + ( 14 J1 J4 − 12 J1 J3 − 2 K18 − 14 J1 J5 + K14 + 1 J −2 J25 ) K1 −J7 K2 − 21 J4 K3 +( 21 J5 + 12 J3 ) K4 +(−3 J7 +J9 ) K5 + 2 6 (J3 + J5 ) K6 + (−K9 − 23 J3 ) K7 + (2 J17 − 21 J2 + 14 J12 ) K9 + J1 K18 + 2 K30 − 2 J9 K21 + 21 J1 K15 , ( 12 J4 − 2 K9 − 2 J3 ) K12 + (−J1 J3 + J6 − J1 K9 − J25 + 2 K15 ) K1 − J4 K3 + (−2 K10 + J4 + K9 ) K4 + (−J7 − J9 ) K5 + J4 K6 − 21 J4 K7 + J17 K9 + ( 12 J12 − 2 J17 − 12 J2 ) K10 + J1 K16 , 2 2 2 2 2 J17 J5 K21 +K17 +J33 − 14 K73 −K12 +9 K25 +4 K33 +9 K13 −3 K21 K19 + 3 3 3 (6 J1 K8 − 5 K21 K9 + 4 J1 K17 + ( 2 J5 + 2 J4 ) K21 − 10 K27 + 32 J1 K25 + 3 J J − 6 J24 + 4 K28 − 32 K33 − 15 J − 72 J12 J17 + 11 J J )K7 + 2 1 19 2 23 2 1 20 (− 12 J20 − J19 ) K24 + (−6 J19 + 32 K17 − 8 K13 + 10 K24 + 4 J1 J17 ) K12 + ( 23 J1 J5 +J25 +2 J26 −2 J1 J3 ) K31 +2 K16 K31 −28 K15 K31 +(10 K13 + 2 6 K25 + 3 K12 + K17 ) K8 + (− 23 J20 + 3 J19 ) K25 − J7 K52 − 3 J8 K21 + 3 11 3 2 (( 2 J17 +2 J1 ) K21 − 2 J18 −10 J1 K31 +12 K32 ) K9 +( 2 J2 +3 J17 ) K33 + (−3 J1 K22 +13 J1 K11 +2 J18 − 12 J12 K21 −6 K32 +3 K23 ) K10 +(4 K18 + 11 J6 − 2 K14 + 21 J1 J5 + 16 K16 − 8 K15 − 3 J1 J3 + 23 J1 J4 ) K11 + ( 23 K17 − 8 K25 + 6 J19 − 6 K24 ) K13 − 6 K32 K4 + ((2 J1 J3 − 6 J1 K10 − K18 + 6 K14 ) K2 + (2 J19 + 6 K12 − J1 K4 ) K3 + 12 J26 + 9 J1 K9 − 29 4 3 K4 K24 +(−2 K16 +2 J26 −6 K15 −3 K14 −J1 J5 ) K5 +(−J13 +3 K17 + 2 K8 ) K6 − K7 K17 − 7 J17 K8 + 13 K22 K9 + 2 J1 K21 K10 + 21 J2 K12 + (J2 +2 J12 ) K13 + 72 J17 K17 −6 J26 K21 +(8 J4 −7 J3 −J5 ) K22 +2 J12 K24 − 3 J K25 −6 J1 K27 +(−11 J4 +20 J3 ) K31 +5 J29 −4 J28 − 23 J1 J22 )K1 + 2 17 (13 K31 − 3 K22 ) K14 − 6 J7 K22 + 6 K42 K6 + 43 J2 K72 + ((9 J5 − 6 J4 + 12 K10 −4 J3 ) K3 +(−15 K9 +4 J4 −J3 −10 J5 ) K4 +(− 59 J + 3 J ) K5 + 4 7 4 8 (17 K10 − 51 K9 ) K6 + 32 K9 K7 + (7 J2 + 3 J17 − 6 J12 ) K9 + (− 15 J + 4 4 2 2 J17 ) K10 + 6 K19 − 10 K30 + 11 J J − 3 J K − 5 J J − 2 J K 1 16 1 25 1 15 + 2 1 26 1 2 J1 K14 )K2 −5 K31 K18 +( 4 J1 J2 +6 J19 −7 K24 +3 K25 ) K17 +(−K42 + 9 J17 K4 − 6 J4 K5 + 4 J12 K6 − 21 J K7 − 7 J1 K12 + J1 K17 + (3 J4 + 4 2 2 + 6 K27 )K3 + (− 21 K17 + 2 J3 + 3 J5 ) K21 − 8 K28 − 32 J23 + 7 K33 − 23 J17 3 J J − 32 J1 J17 ) K13 + (2 K10 K6 + ( 12 J12 − 32 J2 ) K9 − 3 K30 − 4 J10 − 4 1 2 1 J J +J1 K14 −3 K19 −J1 K18 − 12 J12 J5 +3 J2 J3 +6 J1 K15 −4 J2 J4 + 4 2 5 2 J1 J25 − 92 J2 K10 − 4 J1 J6 )K5 + (continued on the next page)
Computation of Affine Covariants of Differential Systems
29
2 (−5 K9 K5 − 34 J2 K7 − 4 K21 K10 − 2 J1 K13 + (−J3 + J4 ) K21 + 23 J17 + 29 1 2 1 1 7 K33 ) K4 + (− 4 K10 K2 + 2 J1 K3 + (− 2 K10 + 8 J3 − 3 J4 − 2 K9 + 3 J5 ) K5 + (−3 J2 − 52 J12 ) K6 + J1 K13 + 2 J3 K21 + 3 K27 + 41 K72 − Q33 2 J J − 3 J K − 3 J K )K 2 + ( 17 J 2 − 3 J2 − 52 K7 ) K62 + (J3 − 2 1 (end) 3 1 20 4 1 17 4 2 7 1 J − 3 J4 ) K23 + (−20 J1 K13 − 7 K21 K10 + 10 K33 + 27 K21 K9 + K72 − 2 5 2 10 K27 − 2 J24 − 5 J17 K7 − J3 K21 + 6 J1 K12 − 7 J1 K8 )K6 − 27 J25 K22 + 8 J17 K27 − 92 J17 K28 + 32 J1 K29 , J1 K30 + 2 K21 K20 + J34 + (−7 J3 − 4 J4 ) K12 + 21 J2 K18 + (J5 + J4 + 5 J3 ) K24 + (2 J8 K2 + (−2 K10 − 2 J3 + 4 J5 ) K3 + (2 J3 − 2 K10 − 6 J5 ) K4 +(− 21 J8 − 12 J7 ) K5 +(3 J5 −J4 ) K6 + 32 K9 K7 +(J17 +J12 ) K9 − 4 J1 K14 −J12 K10 −J17 J3 +2 K19 − 23 J1 K16 +4 J1 K15 )K1 − 21 K13 K9 + 4 K8 K9 + J17 K14 − 5 J17 K15 − 2 J3 K25 − J9 K22 − 2 J13 K21 − J7 K11 + Q34 (2 K15 −5 J26 +10 J1 K10 +10 K16 +J25 ) K3 −3 K20 K2 +(7 J26 −4 J25 + J K10 − 4 J1 K9 − 5 K18 ) K4 + ( 21 K18 − 3 K14 + 10 K14 − 8 K16 − 17 2 1 5 K16 + 2 J1 K9 + J1 J5 ) K12 − J9 K31 + (4 J13 + K20 ) K5 + (−J1 K10 + 1 J J ) K6 +(5 J4 −3 J3 ) K13 +(−2 K24 −J1 J17 +6 K13 ) K10 +(3 J20 + 2 1 5 2 K24 − K17 − 2 K13 − K12 ) K9 + (− 21 K18 − 5 J6 − 12 J1 K9 − J25 ) K7 , , (−11 K12 − 2 K25 − 2 K24 + 17 K13 + 4 K17 ) K9 + 2 J8 K31 + (6 K25 − 7 K12 − J20 − 2 K24 − 20 K17 ) K10 + (−3 J8 + 12 J9 − 6 J7 ) K11 + ( 21 J5 + 4 J3 ) K12 + ( 32 J5 + J4 − 4 J3 ) K13 + (− 12 J12 + 12 J2 + J17 ) K14 + J1 K30 + J5 K24 +(15 J4 +4 J5 +J3 ) K17 +J1 K19 +(−J12 +3 J2 ) K18 +(− 21 J1 J7 + 1 J J ) K21 +(− 32 J5 +2 J3 ) K13 +(J9 −2 J8 ) K22 −6 K21 K20 +( 54 J1 J5 − 2 1 8 4 J6 −3 J1 J3 +J1 J4 ) K12 +(−J4 +J5 ) K25 +(J7 K2 +15 K3 K9 +(10 J3 − Q35 6 J4 −11 K9 ) K4 +8 J9 K5 +(26 J3 −12 K10 −3 J5 −15 J4 +22 K9 ) K6 + (6 K10 + 23 J4 ) K7 + 6 K30 + J1 J25 − J2 K10 − 2 J7 K21 )K1 + (− 47 J1 J8 − 5 K20 − 54 J1 J7 + 12 J12 − 32 J13 + 5 J11 ) K2 + (−3 K14 + J6 + 6 K18 − 3 J K9 +K15 + 34 J1 J5 ) K7 +(−8 K9 +7 K10 ) K8 +(3 J1 K9 +6 J1 K10 + 2 1 7 K16 + J25 − J26 − K14 − 8 K15 ) K4 + (3 J11 − 4 J1 J9 − 9 J12 ) K5 + (6 J6 − 26 K15 − 20 K18 ) K6 + (2 J26 − 3 J25 + K14 − 5 J1 K10 − 6 K15 + J1 K9 ) K3 + J35 , − 12 J8 K12 + (2 K20 − 14 J1 J8 − 14 J1 J9 ) K1 + ( 12 J9 + 12 J7 ) K3 + ( 12 J8 − 2 + (− 12 J5 + 12 J4 − Q36 +( 12 J8 − 12 J7 ) K6 − 12 J7 K7 + (J3 − K10 ) K9 + K10 J3 ) K10 + J36 + 12 J1 K20 .
References [1] H.Derksen, H.Kraft, Constructive Theory, Collection Alg`ebres non commutatives, groupes quatiques et invariants, S´emin. Congr., 2 (1995), SMF, 221 -
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224. ´, J.Carrell, Invariant theory, old and new, Advances in Math. [2] J.Dieudonne 4 (1970), 1-80. [3] G.B.Gurevitch, Foundations of the theory of algebraic invariants, Noordhoff, Ed; Groningen (1964). [4] D. Hilbert, Invariant Theory, 1893 . [5] W.J. Haboush, Reductive groups are geometrically reductive Ann. of Math., 102, 1 (1975, 67 - 83. [6] M.Hochster, J.Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macauley, Advances in Math. 13 (1974), 115-175. [7] V. L. Popov, Invariant Theory, Amer. Math. Soc. Trans. (2), Volume 148 (1991), 99 - 112. [8] V. L. Popov, Constructive Invariant Theory, Ast´erisque 87-88 (1981), 303 334. [9] C.S. Sibirskii, Algebraic invariants of differential equations and matrices, Schtiintsa, Kishinev, Moldova (1976) (en russe). [10] C.S. Sibirskii, Introduction to the algebraic theory of invariants of differential equations, Nonlinear Science, Theory and Applications, Manchester University Press, 1988. [11] N.I. Vulpe, Polynomial bases of comitants of differential systems ans their applications in qualitative theory (russian, eng. and french summ.), “Shtintsa”, Kishinev, 1986, 172 pp.1.60r. [12] N.I. Vulpe, Number of linearly independant comitants of a system of differential equations, Differential equations, 15, 6, 963-973. [13] H. Weyl, The classical groups, their invariants and representations. Princeton 1939. [14] P. A.Worfolk, Zeros of equivariant vector fields : Algorithms for an invariant approach, J. Symbolic Computation, 17 (1994), 487 - 511.
