c Pleiades Publishing, Ltd., 2011.
ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2011, Vol. 32, No. 1, pp. 95–102. GL2(C)
(C)-Orbits of Binary Rational Forms
P. Bibikov* and V. Lychagin**
(Submitted by V. Lychagin)
Institute of Control Sciences RAS, 117997 Moscow, Profsoyuznaya 65 (Russian Federation)
Received October 12, 2010
Abstract—In this paper we study orbits of GL2 (C)-action on the space of binary forms and find
criteria for equivalence of binary forms. The similar results are also obtained for rational binary forms
and for real binary forms.
DOI: 10.1134/S1995080211010069
Keywords and phrases: Binary forms, group action, differential invariant, jet space.
1. INTRODUCTION
Let Vn = S n (C2 )∗ be a space of binary forms of degree n over field C. Consider an action of group
GL2 (C) on space Vn such that subgroup SL2 (C) ⊂ GL2 (C) acts by linear coordinate transformations,
and center C∗ ⊂ GL2 (C) acts by homotheties f → λf , where f ∈ Vn and λ ∈ C∗ . In this paper we give
a solution of classification problem of the action.
This problem is closely connected with other classification problems in invariant theory such that
classification problem of the action of projective group PGL2 (C) on the projective line and with
classification of hyperelliptic curves of genus g (see [1]). The SL2 (C)-invariant algebras of binary forms
of degree n 8 are known so far (see [2–4]).
The case n = 3 was solved by Bool in 1841.
The first nontrivial case n = 4 was solved by Bool, Cayley and Eisenstein in 1841–1850 and initiated
the classical invariant theory (note that the problem of classification of binary forms of degree 4 is closely
related with the cross ratio of the four projective points on projective line, and with j-invariant of elliptic
curve; see [1]).
The case n = 5 was studied by Cayley. In this case the invariant algebra is generated by four
homogeneous polynomials of degrees 4, 8, 12 and 18, which satisfy a homogeneous relation of degree 36
(note that the invariant of degree 18 consists of more than 800 monomials).
The cases n = 6, 7, 8 were studied by Gordan, Shioda, Dixmier and Lazard. The explicit form of
generators of the invariant algebra for n = 7 were found by Bedratyuk in 2007 only (see [4]).
In this paper we suggest the method which classifies GL2 (C)-orbits of binary forms of any degree.
This method is based on the interpretation of space Vn as the space of smooth solutions of differential
equation xux + yuy = nu (the so called Euler equation) and consideration of the GL2 (C)-action on
the solution space of this equation. This interpretation makes it possible to use differential invariants
instead of polynomial ones. The structure of differential invariant algebra is much simpler than the
structure of the classical polynomial invariant algebra, because of Lie-Tresse’s Theorem, which claims
that differential invariant algebra is generated by a finite number of (functionally) independent invariants
and invariant derivations. Under some conditions, this theorem is valid for pseudogroup actions on the
*
**
E-mail: [email protected]
E-mail: [email protected]
The first author was partially supported by the Moebius Contest Foundation for Young Scientists.
95
96
BIBIKOV, LYCHAGIN
differential equations (see [6]). So, to find differential invariant algebra one should find several basic
invariants and invariant derivations.
We begin with more general problem of classification GL2 (C)-orbits on the space C ∞ (C2 ) of
analytic functions. In Section 3 we describe orbits of binary forms. Finally, in Section 4 we give
GL2 (C)- and SL2 (C)-classification of binary rational forms.
2. GL2 (C)-CLASSIFICATION OF ANALYTIC FUNCTIONS
Let C2 be the plane with the coordinates (x, y). Denote by J k C2 spaces of k-jets of analytical
functions with canonical coordinates (x, y, u, u10 , u01 , . . .). Group GL2 (C) acts on the space of analytic
functions C ∞ (C2 ) in following way. Namely, subgroup SL2 (C) ⊂ GL2 (C) acts by linear coordinate
transformations, and center C∗ ⊂ GL2 (C) acts by homotheties f → λf , where f ∈ Vn and λ ∈ C∗ . This
action prolongs to actions in k-jet spaces J k C2 .
By a differential invariant of order k we understand a function I ∈ C ∞ (J k C2 ), which is invariant
with respect to prolonged action of group GL2 (C). We consider only invariants which are analytical in x
and y and polynomial in uσ , u−1 and E −1 (see Proposition 1).
In a similar way one defines an invariant derivation as a linear combination of total derivatives
d
d
d d
+ B dy
(where A, B ∈ C ∞ (J ∞ C2 ) and dx
, dy are the total derivatives), which is invariant
∇ = A dx
with respect to the prolonged action of group GL2 (C).
Note that for such derivations functions ∇(I) are differential invariants (of order, as a rule, higher than
the order of I) for any differential invariant I. This observation allows us to construct new differential
invariants from known ones by differentiations only.
Proposition 1. Function
xu10 + yu01
E=
u
is the differential invariant of order 1, functions
u2 u20 − 2u10 u01 u11 + u210 u02
u20 u02 − u211
and
F = 01
2
u
u3
are differential invariants of order 2, and derivations
d
u10 d
d
u01 d
+y
and
∇2 =
−
∇1 = x
dx
dy
u dx
u dy
H=
are invariant. Moreover, the following equality holds:
E22 − E1 F + HE 2 = 0
(we used notation Ij for ∇j I).
First of all let us describe GL2 (C)-orbits in the spaces of 0-jets, 1-jets and 2-jets. There are two
orbits of incomplete dimension in the space J 0 C2 (namely, orbits {u = 0} and {x = y = 0}). These
orbits will be called singular orbits. In the space J 1 C2 , there are two types of the orbits of incomplete
dimension: orbit {E = 0} and orbits, whose projections in J 0 C2 are singular. In a similar way, the orbits
of incomplete dimension in the space J 2 C2 are orbits lying in {H = 0} and the orbits, whose projections
in J 1 C2 have incomplete dimension. All these orbits are called singular. Finally, GL2 (C)-orbits in
spaces J k C2 are called singular, if their projections in J 2 C2 are singular. It worth to note that regular
orbits in spaces J k C2 have dimension dim GL2 (C) = 4, when k 1.
Theorem 1. The algebra of GL2 (C)-differential invariants is locally generated by invariants
E and F and invariant derivations ∇1 and ∇2 . It separates regular orbits and its syzygies are
generated by following relation:
(E22 − F1 )E − 3E22 + 3E1 F − 4EF = 0.
LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 32 No. 1 2011
GL2 (C)-ORBITS OF BINARY RATIONAL FORMS
97
Proof. Consider a regular orbit O1 ⊂ J 1 C2 . Its codimension is equal to dim J 1 C2 − dim GL2 (C) =
1. On the other hand, we have nonzero invariant E of order 1. Then invariant E generates the invariants
of order 1 and separates regular orbits. In a similar way, codimension of a regular orbit in space J 2 C2
is equal to 4 and is equal to the number of independent invariants of order 2 (these invariants are E,
∇1 E, ∇2 E and F ). Finally, there exist four independent invariants of pure order 3 in the space of 3-jets.
On the other hand, after the derivations of the invariants E and F we get five invariants. Therefor, there
exists a relation among them.
Now let us consider case k 3 and a regular orbit Ok in space J k C2 . Then its projection Ok−1 =
πk,k−1 (Ok ) ⊂ J k−1 C2 is also regular. Dimension of the bundle πk,k−1 : Ok → Ok−1 is equal to
(dim J k C2 − dim GL2 (C)) − (dim J k−1 C2 − dim GL2 (C)) = k + 1.
We claim that using the invariants E and F and derivations ∇1 and ∇2 it is possible to get (k + 1)
independent differential invariants of pure order k.
d
d
and dy
correspondingly. Then the symbols of
Let d1 and d2 be the symbols of the total derivatives dx
2
10 d2
invariant derivations ∇1 and ∇2 equal Ξ1 = xd1 +yd
and Ξ2 = u01 d1 −u
, and the symbols of invariants
u
u
2
E and F equal Ξ1 and Ξ2 correspondingly. Note that symbols Ξ1 and Ξ2 are independent, when E = 0.
So, the symbol of invariant Eσ1 ,σ2 = ∇σ1 1 ∇σ2 2 E is equal to Ξ1σ1 +1 Ξσ2 2 , and symbol of invariant Fτ1 ,τ2 =
∇τ11 ∇τ22 F is equal to Ξτ11 Ξ2τ2 +2 (here σ1 + σ2 = k − 1 and τ1 + τ2 = k − 2). We get, that the vector space
generated by invariants Eσ1 σ2 has dimension k, and the vector space generated by invariants Fτ1 τ2 has
dimension k − 2. On the other hand there exist k − 2 syzygies which are obtained from the syzygy of
Proposition 1 by derivations. So, the intersection of these two spaces has the dimension k − 2, and there
exist k + (k − 1) − (k − 2) = k + 1 independent invariants from the complement of this intersection.
Note that invariants Eσ1 σ2 and Fτ1 τ2 are linear with respect to the leading derivatives, so for any
differential invariant I there exists such polynomial P = P (Eσ1 σ2 , Fτ1 τ2 ), that the invariant I − P has
the order less than I. The induction over the order of invariant completes the proof.
2
∞
2
We say that function f ∈ C (C ) is regular in neighborhood O if dE(f ) ∧ dF (f ) = 0 in O. For
any regular function f one has (locally) the following relations, which have to satisfy the syzygy relation:
⎧
E1 (f ) = F1 (E(f ), H(f )),
⎪
⎪
⎪
⎨
E2 (f ) = F2 (E(f ), H(f )),
⎪
H1 (f ) = F3 (E(f ), H(f )),
⎪
⎪
⎩
H2 (f ) = F4 (E(f ), H(f )).
Note also that the syzygy relation is a compatibility condition of this PDE system.
Theorem 2. Functions (F1 , F2 , F3 , F4 ) satisfying the syzygy relation (locally) determine the
GL2 (C)-orbit of regular function f ∈ C ∞ (C2 ).
Proof. It is obvious that if functions f1 and f2 are equivalent in neighborhood O, then the corresponding dependencies coincide.
Let us prove the converse statement. Note that any regular orbit is a connected set of dimension 4.
Our system is integrable (because of the syzygy condition) and has finite type, so its solution space also
has dimension 4. So, if two functions have the same set of dependencies, their graphs in 5-jet space
2
J 5 C2 are locally equivalent. Then these functions are also locally equivalent.
3. GL2 (C)-ORBITS OF BINARY FORMS
In this section we consider the GL2 (C)-action on space Vn of binary forms.
Recall that space Vn of binary forms of degree n we are identified with the space of smooth solutions
of the Euler equation xux + yuy = nu. The corresponding algebraic manifold E ⊂ J 1 C2 is given by
equation xu10 + yu01 = nu.
Proposition 2. Prolongations E (k) ⊂ J k C2 , k = 1, 2, . . . of the Euler equation are given by
equations
xui,m−i + yui−1,m−i+1 = (n − m + 1)ui−1,m−i ,
LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 32 No. 1 2011
98
BIBIKOV, LYCHAGIN
where i = 1, . . . , m, m = 1, . . . , k and u00 = u.
By a differential invariant of order k of binary form we mean a GL2 (C)-invariant function on the
manifold E (k) , which is polynomial in uσ and u−1 .
Theorem 3. The differential invariants algebra of the GL2 (C)-action on the manifold E (∞) is
freely generated by differential invariant H and invariant derivation ∇ = ∇2 |E (∞) .
Proof. Note that restrictions of invariants F and H on the equation E (2) satisfy the following relation:
F = n(n − 1)H. Consider a regular orbit Ok+1 ⊂ E (k+1) . The dimension of bundle πk+1,k : Ok+1 → Ok
is equal to
(dim E (k+1) − dim GL2 (C)) − (dim E (k) − dim GL2 (C)) = 1.
On the other hand, the number of independent invariants of order k + 1 is one more than the number of
invariants of order k.
2
Now consider invariants I1 = H, I2 = ∇H and I3 = ∇2 H. The restrictions of these invariants
on graph L4f ⊂ J 4 C2 of form f ∈ Vn are homogeneous rational functions in x and y. Then they are
algebraically dependent and F (I1 (f ), I2 (f ), I3 (f )) = 0 for some irreducible polynomial F . Let us order
variables Ik by requirement that I1 ≺ I2 ≺ I3 and assume that polynomial F has the minimal degree
with respect to this order and is defined up to non zero scalar.
Definition. We say that binary form f ∈ Vn is regular, if
(2I1 I3 − 3I22 )(f ) = 0.
Theorem 4. Let f1 , f2 ∈ Vn be binary forms and F1 , F2 be the corresponding dependencies
between invariants Ik . Then forms f1 and f2 are GL2 (C)-equivalent if and only if F1 = F2 .
Proof. It is obvious that for equivalent forms f1 and f2 , one has F1 = F2 .
Let us prove the converse statement. Consider an arbitrary regular binary form f ∈ Vn . It’s enough to
prove that the orbit O of the graph L4f ⊂ E (4) of regular form f coincides with the irreducible component
of algebraic manifold E (4) ∩ {F = 0} (then the required statement follows from the Theorem of existence
and uniqueness solutions for completely integrable systems). It is obvious that Ō ⊆ E (4) ∩ {F = 0}. It
follows from Proposition 2 that dim(E (4) ∩ {F = 0}) = 6. Let us prove that dim Ō = 6. It’s enough to
show that 4-prolongations of vector fields from Lie algebra gl2 (C) are transversal to the graph L4f of form
f . The straightforward computation shows that this condition may be written as (2I1 I3 − 3I22 )(f ) = 0.
It’s equivalent to the condition of regularity of the form f . Therefore, if the form f is regular, then the
prolongations of vector fields from gl2 (C) are transversal to the graph L4f and dim Ō = 6.
Thus, dim Ō = 6 = dim(E (4) ∩ {F = 0}). Moreover, Ō is the irreducible algebraic manifold defined
by the system of equations Ik = Ik (f ), k = 1, 2, 3. Therefore, Ō coincides with the irreducible component of the manifold E (4) ∩ {F = 0}.
So, it’s enough to prove that the graph L4f of any regular form f belongs to the fix irreducible
component of manifold E (4) ∩ {F = 0}. This statement is equivalent to the irreducibility of polynomial
F (I1 , I2 , I3 ) = FJ = F (x, y, u, . . .) corresponding f . Assume the converse. Let FJ = A · B be a decomposition of polynomial FJ into mutually-prime multipliers. Then for each vector field v ∈ sl2 (C) we
have
v (4) (A) · B + A · v (4) (B) = 0.
So, the polynomial v (4) (A) is divided by A and v (4) (A) = λv A, where λv is a polynomial. Compare
degrees of left and right parts, we get that λv ∈ C is a constant. Thus λ defines 1-cocycle c : sl2 (C) → C,
c(v) = λv . But all 1-cocycles of the Lie algebra are trivial, so λv = 0. Finally, we get that v (4) (A) = 0
for all v ∈ sl2 (C). It implies that the polynomial A is sl2 (C)- and gl2 (C)-invariant. Using Theorem 3,
we have A = A(I1 , I2 , I3 ). Then the polynomial F is reducible. This contradiction completes the proof
of the regular case.
Let now f ∈ Vn a non-regular form, i.e. (2I1 I3 − 3I22 )(f ) = 0. This relation implies the existence
of algebraic dependence between polynomials I1 (f ) and I2 (f ). Obviously, that this dependence has
LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 32 No. 1 2011
GL2 (C)-ORBITS OF BINARY RATIONAL FORMS
99
ξ2
10
8
6
4
2
0
4
2
6
8
10
ξ1
Fig. 1. Graph of the function F (1, ξ1 , ξ2 ) = 0 for the form f (x, y) = xy(x + y)(−2x + y), where ξ1 = I22 /I13 and
ξ2 = I3 /I12 .
ξ2
10
–10
–5
5
10
ξ1
–10
–20
–30
(x+y)(x+2y)
Fig. 2. Graph of the function F (1, ξ1 , ξ2 ) = 0 for the form f (x, y) = xy (x−y)(x−2y)
, where ξ1 = I22 /I13 and ξ2 =
I3 /I12 .
the form F (I1 , I2 ) = I13 − λI22 . If I1 (f ) = 0, then dimension of the graph’s L3f orbit is equal to 5. If
I1 (f ) = 0, then the dependence has the form F (I1 ) = I1 − μ, and the dimension of the graph’s L2f
orbit is equal to 4. It is easy to prove, that in the first case the orbit of the graph L3f coincides with
the manifold E (3) ∩ {F = 0}, and in the second case, that the orbit of the graph L2f coincides with the
manifold E (2) ∩ {F = 0}. The same argumentation as in the regular case completes the proof.
2
Remark. It can be proved that the singularity condition (2I1 I3 − 3I22 )(f ) = 0 is equivalent to the
condition that the form f ∈ Vn has no more than two roots without taking multiplicity, or that the form
f is equivalent to the form xk y n−k for some k n, or that the orbit of form f has the dimension less
than 4.
Examples. (1) Consider forms f1 (x, y) = xy(x + y)(−2x + y) and f2 (x, y) = xy(x + y)(−3x + y).
LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 32 No. 1 2011
100
BIBIKOV, LYCHAGIN
The corresponding dependencies between invariants Ik are
F1 (I1 , I2 , I3 ) = 3087I24 − (12348I3 + 16464I12 )I1 I22 − (800I33 − 7548I22 I32 − 23328I14 I3 − 15552I16 ),
F2 (I1 , I2 , I3 ) = 19773I24 − (79092I3 + 105456I12 )I1 I22 − (9800I33 − 20292I22 I32
− 93312I14 I3 − 62208I16 ).
Then F1 = F2 and the forms f1 and f2 do not belong to the same orbit. From the point of view of classical
invariant theory orbits of binary forms of degree 4 are determined by j-invariant (see [1]). In our case j133
invariant of the first form equals 343
36 and of the second one— 144 .
(2) Consider forms f1 (x, y) = xy(x + y)(−x + y) and f2 (x, y) = xy(x + y)(2x + y). The corresponding dependencies between invariants Ik are
F1 (I1 , I2 , I3 ) = F2 (I1 , I2 , I3 ) = 3I22 − 6I1 I3 − 8I13 .
Therefor, these forms belong to the same orbit. In this case j-invariants of forms equal
27
4 .
4. BINARY RATIONAL FORMS
The results obtained above may be generalized on some other actions. We will consider two examples:
GL2 (C)-action on the space Rn of binary rational forms and SL2 (C)-action on the space Rn .
4.1. GL2 (C)-Orbits of Binary Rational Forms
Let Rn be a space of binary rational forms of degree n (i.e. the difference between the degrees of
numerator and denominator equals n) with he above GL2 (C)-action. All above proofs valid for this
action.
Theorem 5. (1) Let f1 , f2 ∈ Rn be rational forms of degree n = 0, 1 and F1 , F2 are the
corresponding dependencies between the invariants Ik . Then the forms f1 and f2 are equivalent if
and only if F1 = F2 .
(2) Rational forms f1 and f2 of degree 1 are equivalent if and only if the forms f1−1 and f2−1 of
degree −1 are equivalent.
(3) Rational forms f = f1 /f2 and g = g1 /g2 of degree 0 are equivalent if and only if binary
forms λ1 f1 + λ2 f2 are λ1 g1 + λ2 g2 are equivalent (here λ1 , λ2 are considered as the formal
variables).
Note that the classification of orbits of binary rational forms is equivalent to classification of the
ordered pairs (f, g) of binary forms, and Theorem 5 classify pairs of binary forms.
4.2. SL2 (C)-Orbits of Binary Rational Forms
Finally, we consider SL2 (C)-action on space Rn . The main difference between this action and the
GL2 (C)-action is the existence of the invariant U = u of degree 0. Then the differential invariant algebra
is generated by invariants U and H0 = u2 H and derivation ∇0 = u∇.
Consider invariants J1 = U , J2 = H0 and J3 = ∇0 H0 and algebraic dependency between their values
on a form f ∈ Rn : F (J1 (f ), J2 (f ), J3 (f )) = 0.
Theorem 6. (1) Let f1 , f2 ∈ Rn be rational forms of degree n = 0, 1 and F1 , F2 are the
corresponding dependencies between the invariants Jk . Then this forms are SL2 (C)-equivalent
if and only if F1 = F2 . (2) Rational forms f1 and f2 of degree 1 are equivalent if and only if the
forms f1−1 and f2−1 of degree −1 are equivalent.
(3) Rational forms f = f1 /f2 and g = g1 /g2 of degree 0 are equivalent if and only if the
binary forms λ1 f1 + λ2 f2 are λ1 g1 + λ2 g2 are equivalent (here λ1 , λ2 are considered as the formal
variables).
As in Theorem 5, this theorem makes it possible to classify SL2 (C)-orbits of pairs of binary forms.
Finally, let us note that Theorem 6 can be generalized for the case of real numbers. Namely, the
following theorem holds.
LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 32 No. 1 2011
GL2 (C)-ORBITS OF BINARY RATIONAL FORMS
101
Theorem 7. (1) Let f1 , f2 ∈ Rn be rational real forms of degree n = 0, 1 and F1 , F2 are the
corresponding dependencies between the invariants Ik . Then the forms f1 and f2 are equivalent
if an only if F1 = F2 and 3-jets j3 (f1 ) and j3 (f2 ) belong to the same irreducible component of the
manifold E (3) ∩ {F1 = 0}.
(2) Rational real forms f1 and f2 of degree 1 are equivalent if and only if the forms f1−1 and
−1
f2 of degree −1 are equivalent.
(3) Rational real forms f = f1 /f2 and g = g1 /g2 of degree 0 are equivalent if and only if the
binary forms λ1 f1 + λ2 f2 are λ1 g1 + λ2 g2 are equivalent (here λ1 , λ2 are considered as the formal
variables).
APPENDIX
In this section we give a computer program in Maple which solves the question “are two given
regular rational forms f and g equivalent?”. Here ξ1 = I22 /I13 and ξ2 = I3 /I12 , and polynomial F is
presented as polynomial in ξ1 and ξ2 .
>with(DifferentialGeometry):with(JetCalculus):with(Tools):with(PDETools):
>with(LinearAlgebra):with(PolynomialIdeals):with(plots,implicitplot):
>Preferences(<<JetNotation>>,<<JetNotation2>>):DGsetup([x,y],[u],Bin,4):
>Tools:-DGinfo(Bin,<<FrameJetVariables>>):
>
# INVARIANTS AND INVARIANT DERIVATIVES
>H:=u[20]*u[02]-u[11]^2:
>S:=proc(f) simplify(u[01]*TotalDiff(f,x)-u[10]*TotalDiff(f,y)) end proc:
>L:=proc(h) {u[00]=h, u[10]=diff(h,x), u[01]=diff(h,y), u[20]=diff(h,x,x),
u[11]=diff(h,x,y), u[02]=diff(h,y,y), u[30]=diff(h,x,x), u[21]=diff(h,x,x,y),
u[12]=diff(x,y,y), u[03]=diff(h,y,y,y), u[40]=diff(h,x,x,x,x), u[31]=diff(h,x,x,x,y),
u[22]=diff(h,x,x,y,y), u[13]=diff(h,x,y,y,y), u[04]=diff(h,y,y,y,y)} end proc:
>Hess:=proc(f) simplify(eval(H, L(f))) end proc:
>
# INVARIANT OF RATIONAL BINARY FORM
>First:=proc(f) simplify(eval(eval(S(H), L(f))^2/eval(H, L(f))^3, y-t*x)) end proc:
>Second:=proc(f) simplify(eval(eval(S(S(H)), L(f))/eval(H, L(f))^2, y-t*x)) end proc:
>Invariant:=proc(f) local P, Q, R, T, I;
P:=numer(Second(f))-xi[2]*denom(Second(f)); Q:=numer(First(f))-xi[1]*denom(First(f));
R:=factor(resultant(P, Q, t));
T:=simplify(Radical(‘<,>‘(R)))
I:=Generators(S); I[1];
end proc
>
# EQUIVALENCE OF TWO BINARY FORMS
>equivalence:=proc(f,g) local J;
J:=‘<,>‘(Invariant(f));
IdealMembership(Invariant(g), J)
end proc
Below we give an example of application of this program.
>f:=(x^4+y^4)/(x^2+y^2):
>g:=(2*x^4+4*x^3*y+30*x^2*y^2+28*x*y^3+17*y^4)/(2*x^2+2*x*y+5*y^2):
>equivalence(f,g);
true
>Invariant(f);
-147456*xi[1]^2-36864*xi[2]^2*xi[1]+165888*xi[2]*xi[1]^2-28032*xi[2]^3*xi[1]+
+178848*xi[2]^2*xi[1]^2+90396*xi[2]*xi[1]^3+2808*xi[2]^4*xi[1]-15552*xi[2]^3*xi[1]^2-1236384*xi[1]^3-2304*xi[2]^4-2187*xi[1]^4-2208*xi[2]^5+200*xi[2]^6
LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 32 No. 1 2011
102
BIBIKOV, LYCHAGIN
REFERENCES
1.
2.
3.
4.
E. Vinberg and V. Popov, Invariant theory, VINITI 55 (1989).
J. Dixmier, Quelques aspects de la théorie des invariants, Gaz. Math., Soc. Math. Fr. 43, 39 (1990).
P. Gordan, Invariantentheorie (Teubner, Leipzig, 1885).
L. Bedratyuk, On complete system of invariants for the binary form of degree 7, Journal of Symbolic
Computation 42, 935 (2007).
5. D. Alekseevskii, A. Vinogradov, and V. Lychagin, Basic ideas and concepts of differential geometry,
VINITI 28 (1988).
6. B. Kruglikov and V. Lychagin, Invariants of pseudogroup actions: homological methods and finiteness
theorem, Int. J. Geom. Methods Mod. Phys. 3 (5-6), 1131 (2006).
7. P. Olver, Classical invariant theory (Cambridge Univercity Press, Cambridge, 1999).
LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 32 No. 1 2011