Affine Differential Invariants for Planar Curves

Mehdi Nadjafikhah

Abstract

In this paper we solve the affine equivalence problem for the graph of functions with real values by finding a complete system of differential invariants for the affine group action.

Mathematics Subject Classification: 53A55, 53A15, 58A15. Key words: differential invariant, differential operator, moving frame, moving coframe, normalization

1 Introduction

The differential invariants associated with a transformation group on a manifold are the fundamental building blocks for understanding the geometry, equivalence, symmetry and other properties of submanifolds. The basic theory of differential invariants dates back to the work of Lie, [5] and Tresse, [7]. However, a complete classification of differential invariants for many of the fundamental transformation groups of physical and geometrical importance remains undeveloped.

Cartan, [1] and [2], proved how his moving frame method could produce the differential invariants for several interesting examples. A new, practical approach to the method of moving frames was recently developed by Fels and Olver, [3] and [4]. The method enables one to algorithmically implement both the practical and theoretical construction of moving frames.

In this paper we solve the following problem, by implementing the Fels and Olver’s moving coframe method: Problem. Given two differentiable ordinary functions y = f(x) and y = g(x) and a number x_{0} ∈ D_{f} ∩ D_{g}, does there exist an affine transformation T ∈ A(2) and a positive number ε such that

T · f(x) = g(T · x) for all x ∈ (x_{0} − ε, x_{0} + ε)

According to the Theorem 7.7 of Fels and Olver, [3], we only need to find a complete set of differential invariants. And, by implementing the Fels and Olver moving coframe method, we prove that Theorem. There exists two differential invariants

Balkan Journal of Geometry and Its Applications, Vol.7, No.1, 2002, pp. 69-78.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

I = \frac{sgn(y′′′)\sqrt{3y′′y^{(4)} − 5y^{′′′2}}}{|y′′|}
J = \frac{sgn(y′′y′′′)(9y^{′′2}y^{(5)} − 45y′′y′′′y^{(4)} + 40y^{′′′3})}{(\sqrt{3y′′y^{(4)} − 5y^{′′′2}})^{3}}

of order 4. Furthermore, for any k ≥ 0, a complete system of functionally independent differential invariants of order k + 4 is provided by I, J, DJ, D^{2}J, ... , and D^{k}J,

where D = (D_{x})^{−1}D_{x} is the associated differential operator and D_{x} = \frac{∂}{∂x} + y′ \frac{∂}{∂y} +

y′′ \frac{∂}{∂y′} + ⋯ is the total derivative operator.

2 Formulation the problem

If A(2) = GL(2) ∝ R^{2} is the affine transformations group in the plane, then we can identify A(2) ⊆ GL(3) as a subgroup of GL(3) by identifying (R, a) ∈ A(2) with the 3 × 3 matrix

( \begin{matrix}R & a \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}a & b & x_{0} \\ c & d & y_{0} \\ 0 & 0 & 1 \end{matrix} ) , ad − bc ≠ 0 (1)

then, the six independent Maurer-Cartan forms are the components of the matrix

μ := ( \begin{matrix}R & a \\ 0 & 1 \end{matrix} )^{−1} · d ( \begin{matrix}R & a \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}R^{−1}dR & a^{−1}da \\ 0 & 0 \end{matrix} ) (2)

That is,

\begin{matrix}μ_{1} = \frac{d da − b dc}{ad − bc} & μ_{2} = \frac{d db − b dd}{ad − bc} \\ μ_{3} = \frac{d dx_{0} − b dy_{0}}{ad − bc} & μ_{4} = \frac{a dc − c da}{ad − bc} \\ μ_{5} = \frac{a dd − c db}{ad − bc} & μ_{6} = \frac{a dy_{0} − c dx_{0}}{ad − bc} \end{matrix} (3)

We define the action of A(2) on R^{2} by

(4)  ( x̄ ; ȳ ; 1 ) = ( \begin{matrix}R & a \\ 0 & 1 \end{matrix} ) · ( \begin{matrix}x \\ y \\ 1 \end{matrix} )

= ( \begin{matrix}ax + by + x0 \\ cx + dy + y0 \\ 1 \end{matrix} ) , ∀ ( \begin{matrix}x \\ y \\ 1 \end{matrix} ) ≡ ( \begin{matrix}x \\ y \end{matrix} ) ∈ R^{2}

as usual. This action is effective and transitive.

3 Finding the most general compatible lift

Let the origin of R^{2}, z_{0} := (0, 0) be the base of problem. In order to compute the most general compatible lift ρ : R^{2} → A(2) with the base z_{0}, we solve the system of 2 equations ρ(z) · z_{0} = z for 2 of group parameters in terms of the coordinates z on R^{2} and 4 remaining group parameters,

x_{0} = x , y_{0} = y (5)

The most general compatible lift thus has the form

ρ_{0}(x, y, a, b, c, d) = ( \begin{matrix}a & b & x \\ c & d & y \\ 0 & 0 & 1 \end{matrix} ) (6)

where a, b, c, and d are arbitrary functions of x and y, subject only to the condition ad − bc ≠ 0, so that the determinant of (6) does not vanish, and hence ρ_{0} does take its values in the group A(2).

By substituting the formulae (5) characterizing our compatible lift (6) into the Maurer- Cartan forms (3), we obtain the moving coframe of order zero

\begin{matrix}μ_{1}^{(0)} = \frac{d da − b dc}{ad − bc} & μ_{2}^{(0)} = \frac{d db−b dd}{ad−bc} \\ μ_{3}^{(0)} = \frac{d dx − b dy}{ad − bc} & μ_{4}^{(0)} = \frac{a dc−c da}{ad−bc} \\ μ_{5}^{(0)} = \frac{a dd − c db}{ad − bc} & μ_{6}^{(0)} = \frac{a dy−c dx}{ad−bc} \end{matrix} (7)

which forms a basis for the space of one-forms on B_{0} = M × H, where H = A(3)_{z_{0}} ≃ GL(2) is the isotropy group of the base point z_{0}.

4 First normalization

Let us now consider a curve N ⊂ M. For simplicity, we shall assume that the curve concedes with the graph of a function y = y(x). We restrict the moving coframe forms to the curve, which amounts to replacing the differential dy its ”horizontal” component y′ dx. Therefore, the restricted (or horizontal) moving coframe forms are explicitly given by

\begin{matrix}μ_{1,h}^{(0)} = μ_{1}^{(0)} & μ_{2,h}^{(0)} = μ_{2}^{(0)} \\ μ_{3,h}^{(0)} = \frac{d−by′}{ad−bc}dx & μ_{4,h}^{(0)} = μ_{4}^{(0)} \\ μ_{5,h}^{(0)} = μ_{5}^{(0)} & μ_{6,h}^{(0)} = \frac{ay′−c}{ad−bc}dx \end{matrix} (8)

which now depend on first order derivatives, and thus are defined on the first jet space J^{1}R^{2} ≃ R^{3}.

The next step in the procedure is to look for invariant combinations of coordinates and group parameters. Among the restricted one-forms (8), there is one linear dependency, namely μ_{3} = J μ_{6}, where J = c + d − (a + b)y′ is a lifted invariant. We can normalize this lifted invariant by setting it equal to one. By solving the equation J = 1 for c, we have c = −d + (a + b)y′ (9)

Substituting (9) into (3) , we find the first order moving coframe

μ_{1}^{(1)} = \frac{da}{a + b} + \frac{by′ db}{(−d + by′)(b + a)}

−\frac{b dd}{(−d + by′)(b + a)} + \frac{a dy′}{−d + by′}

μ_{2}^{(1)} = −\frac{d db}{(−d + by′)(b + a)} + \frac{b dd}{(−d + by′)(b + a)}

(10)  μ_{3}^{(1)} = −\frac{d dx}{(−d + by′)(b + a)} + \frac{b dy}{(−d + by′)(b + a)}

μ_{4}^{(1)} = \frac{da}{a + b} − \frac{ay′ db}{(−d + by′)(b + a)} +

+\frac{a dd}{(−d + by′)(b + a)} + \frac{b dy′}{−d + by′}

μ_{5}^{(1)} = \frac{(−d + by′ + ay′) db}{(−d + by′)(b + a)} − \frac{a dd}{(−d + by′)(b + a)}

μ_{6}^{(1)} = \frac{(−d + by′ + ay′) dx}{(−d + by′)(b + a)} − \frac{a dy}{(−d + by′)(b + a)}

which completely characterizes the group transformations on B_{1}.

5 Second normalization

As before, we determine new lifted invariants by restricting the first order moving coframe one-forms to the curve y = y(x). This amounts to restricting dy and dy′ by their horizontal components y′dx and y′′dx respectively, leading to the restricted forms

μ_{1,h}^{(1)} = \frac{da}{a + b} + \frac{by′ db}{(−d + by′)(b + a)}

−\frac{b dd}{(−d + by′)(b + a)} + \frac{by′′ dx}{−d + by′}

μ_{2,h}^{(1)} = −\frac{d db}{(−d + by′)(b + a)} + \frac{b dd}{(−d + by′)(b + a)}

(11)  μ_{3,h}^{(1)} = μ_{6,h}^{(1)} = \frac{dx}{a + b}
μ_{4,h}^{(1)} = \frac{da}{a + b} − \frac{ay′ db}{(−d + by′)(b + a)}

+\frac{a dd}{(−d + by′)(b + a)} − \frac{ay′′ dx}{−d + by′}

μ_{5,h}^{(1)} = \frac{(−d + by′ + ay′) db}{(−d + by′)(b + a)} − \frac{a dd}{(−d + by′)(b + a)}

Then we have two linear dependencies, namely μ_{5,h}^{(1)} = μ_{1,h}^{(1)} + μ_{2,h}^{(1)} + Kμ_{3,h}^{(1)} − μ_{4,h}^{(1)}, μ_{6,h}^{(1)} = μ_{3,h}^{(1)},

where K = \frac{y′′(a + b)^{2}}{d − by′} is a new lifted invariant. Again, we can normalize this lifted invariant by setting it equal to one. By solving the equation K = 1 for d, we have d = (a + b)^{2}y′′ + by′ (12) Note that we can not normalize K = 0 since this would require a = −b, but then all the one-forms in (10) would have zero denominator. Substituting (12) into (10), we find the second order moving coframe

μ_{1}^{(2)} = \frac{(3b + a) da}{(a + b)^{2}} + \frac{2b db}{(a + b)^{2}} −

−\frac{ab dy′}{y′′(a + b)^{3}} + \frac{b dy′′}{y′′(a + b)}

μ_{2}^{(2)} = \frac{−2b da}{(a + b)^{2}} + \frac{(a − b) db}{(a + b)^{2}} −

−\frac{b^{2} dy′}{y′′(a + b)^{3}} − \frac{b dy′′}{y′′(a + b)}

(13)  μ_{3}^{(2)} = \frac{(y′′′(a + b)^{2} + by′) dx}{y′′(a + b)^{3}} − \frac{b dy}{y′′(a + b)^{3}}
μ_{4}^{(2)} = −\frac{(a − b) da}{(a + b)^{2}} − \frac{2a db}{(a + b)^{2}} +

+\frac{a^{2} dy′}{y′′(a + b)^{3}} − \frac{a dy′′}{y′′(a + b)}

μ_{5}^{(2)} = \frac{2a da}{(a + b)^{2}} + \frac{(3b + a) db}{(a + b)^{2}} −

−\frac{ab dy′}{y′′(a + b)^{3}} + \frac{a dy′′}{y′′(a + b)}

μ_{6}^{(2)} = \frac{(y′′′(a + b)^{2} − ay′) dx}{y′′(a + b)^{3}} − \frac{a dy}{y′′(a + b)^{3}}

6 Third normalization

As before, we determine new lifted invariants by restricting the second order moving coframe one-forms to the curve y = y(x). This amounts to restricting dy , dy′and dy′′ by their horizontal components y′dx, y′′dx and y′′′dx respectively, leading to the restricted forms

μ_{1,h}^{(2)} = \frac{(3b + a) da}{(a + b)^{2}} + \frac{2b db}{(a + b)^{2}} + \frac{b(y′′′(a + b)^{2} − ay′′) dx}{y′′(a + b)^{3}}

μ_{2,h}^{(2)} = \frac{−2b da}{(a + b)^{2}} + \frac{(a − b) db}{(a + b)^{2}} − \frac{b(y′′′(a + b)^{2} + by′′) dx}{y′′(a + b)^{3}}

(14)  μ_{3,h}^{(2)} = μ_{6,h}^{(2)} = \frac{dx}{b + a}

μ_{4,h}^{(2)} = −\frac{(a − b) da}{(a + b)^{2}} − \frac{2a db}{(a + b)^{2}} − \frac{a(y′′′(a + b)^{2} − ay′′) dx}{y′′(a + b)^{3}}

μ_{5,h}^{(2)} = \frac{2a da}{(a + b)^{2}} + \frac{(3b + a) db}{(a + b)^{2}} + \frac{a(y′′′(a + b)^{2} + bay′′) dx}{y′′(a + b)^{3}}

There exist three linear dependencies, namely

μ_{4,h}^{(2)} = −2 μ_{2,h}^{(2)} + L_{1} μ_{3,h}^{(2)}
μ_{5,h}^{(2)} = 3 μ_{2,h}^{(2)} + L_{2} μ_{3,h}^{(2)} , μ_{6,h}^{(2)} = μ_{3,h}^{(2)}

where

L_{1} = \frac{(a + b)^{2}y′′′ + 3by′′}{(a + b)y′′} , L_{2} = −\frac{(a + b)^{2}y′′′ + (2b − a)y′′}{(a + b)y′′}

are new lifted invariants. Again, we can normalize these lifted invariants by setting L_{1} = 1. By solving the equation L_{1} = 1 for a, we have

a = −b + \frac{\sqrt{−3by′′y′′′}}{y′′′} (15)

and then L_{2} = 0. Substituting (15) into (13), we find the third order moving coframe

μ_{1}^{(3)} = \frac{db}{2b} + \frac{\sqrt{3}y′′′(by′′′ − \sqrt{3}\sqrt{−by′′y′′′}) dy′}{9y^{′′2}\sqrt{−by′′y′′′}} +

+\frac{\sqrt{3}y′′′(4by′′′ + \sqrt{3}\sqrt{−by′′y′′′}) dy′}{6y′′\sqrt{−by′′y′′′}} −

−\frac{\sqrt{3}y′′′(2by′′′ + \sqrt{3}\sqrt{−by′′y′′′}) dy′′}{6y′′\sqrt{−by′′y′′′}}

μ_{2}^{(3)} = −\frac{\sqrt{3}y′′′(y′y′′′ − 3y^{′′2}) db}{9y^{′′2}\sqrt{−by′′y′′′}}

(16)  μ_{3}^{(3)} = −\frac{\sqrt{3}y′′′ (y′y′′′ − 3y^{′′2}) dx}{9y^{′′2}\sqrt{−by′′y′′′}} + \frac{\sqrt{3}y′′′^{2} dy}{9y^{′′2}\sqrt{−by′′y′′′}}

μ_{4}^{(3)} = \frac{\sqrt{3}ab}{18y^{′′2}\sqrt{−by′′y′′′}} ((2by′′′(y′y′′′ − 3y^{′′2})) +

+(3y^{′′2} − 2y′y′′′)\sqrt{3}\sqrt{−by′′y′′′}) db +

+\frac{(by′′′ − \sqrt{3}\sqrt{−by′′y′′′}) dy′}{3by^{′′2}} + \frac{dy′′}{2y′′} − \frac{dy′′′}{2y′′′}

μ_{5}^{(3)} = −\frac{\sqrt{3}y′′′ (3by^{′′2} − by′y′′′ + y′\sqrt{3}\sqrt{−by′′y′′′}) db}{9y^{′′2}\sqrt{−by′′y′′′}}

μ_{6}^{(3)} = −\frac{\sqrt{3}y′′′ (3by^{′′2} − by′y′′′ + y′\sqrt{3}\sqrt{−by′′y′′′}) dx}{9y^{′′2}\sqrt{−by′′y′′′}}

7 Fourth normalization

As before, we determine new lifted invariants by restricting the third order moving coframe one-forms to the curve y = y(x). This amounts to restricting dy , dy′, dy′′ and dy′′′ by their horizontal components y′dx, y′′dx, y′′′dx and y^{(4)}dx respectively, leading to the restricted forms

μ_{1,h}^{(3)} = \frac{db}{2b} + \frac{\sqrt{3}(5y^{′′′2} − 3y^{(4)}y′′)(2by′′′ + \sqrt{3}\sqrt{−by′′y′′′}) dx}{18y′′y′′′\sqrt{−by′′y′′′}}

μ_{2,h}^{(3)} = μ_{2}^{(3)}, μ_{3,h}^{(3)} = μ_{6,h}^{(3)} = \frac{\sqrt{3}y′′′ dx}{\sqrt{−by′′y′′′}}
μ_{4,h}^{(3)} = −2 μ_{2,h}^{(2)} + μ_{3,h}^{(2)}, μ_{5,h}^{(3)} = 3 μ_{2,h}^{(2)}

There are four linear dependencies, namely

μ_{3,h}^{(3)} = L_{1}μ_{2,h}^{(3)}, μ_{4,h}^{(3)} = −μ_{1,h}^{(3)} + L_{2}μ_{2,h}^{(3)}
μ_{6,h}^{(3)} = L1μ_{2,h}^{(3)}, μ_{5,h}^{(3)} = 2μ_{1,h}^{(3)} + 3μ_{2,h}^{(3)}
L_{1} = \frac{3y′′y′′′}{b(3y′′y^{(4)} − 5y^{′′′2})}
L_{2} = \frac{10by^{′′′2} − 6by′′y^{(4)} + 3y′′y′′′}{b(3y′′y^{(4)} − 5y^{′′′2})}

where all of the coefficients are lifted invariants. Again, we can normalize the L_{1} setting it equal to one. By solving this equation for b, we have

b = \frac{y′′′}{y^{(4)}} − \frac{3y′′}{5y′′′} (17)

and therefore L_{2} = −1.

Substituting (5), (9), (12), (15) and ( 17) in (6) we find the forth order moving frame

(18)  ρ_{4} : X^{(4)} ⟼ ( \begin{matrix}a & b & x \\ c & d & y \\ 0 & 0 & 1 \end{matrix} )

= ( \begin{matrix}−b + ^{\frac{\sqrt{−3by′′y′′′}}{y′′′}} & \frac{y′′′}{y^{(4)}} − \frac{3y′′}{5y′′′} & x \\ −d + (a + b)y′ & (a + b)^{2}y′′ + by′ & y \\ 0 & 0 & 1 \end{matrix} )

on the fourth jet space J^{4} = J^{4}R^{2} ≃ R^{6} of curves in R^{2} with coordinates X^{(4)} := (x, y, y′, y′′, y′′′, y^{(4)}).

Substituting (17) into (16), we find the fourth order moving coframe

μ_{1}^{(4)} = \frac{y′′′(3Ry′′y^{(4)} − y′′y^{′′′2} − 5Ry^{′′′2})}{3Ry^{′′2}(3y′′y^{(4)} − 5y^{′′′2})} dy′ +

+\frac{−10Ry^{′′′2} + 4y′′y^{′′′2} + 3Ry′′y^{(4)}}{2Ry′′(3y′′y^{(4)} − 5y^{′′′2})} dy′′ −

−\frac{y′′′(−5R + y′′)}{R(3y′′y^{(4)} − 5y^{′′′2})} dy′′′

μ_{2}^{(4)} = \frac{y^{′′′3}}{3Ry′′(3y′′y^{(4)} − 5y^{′′′2})} dy′ − 2\frac{y^{′′′2}}{R(3y′′y^{(4)} − 5y^{′′′2})} dy′′ +

+\frac{y′′y′′′}{R(3y′′y^{(4)} − 5y^{′′′2})} dy′′′

μ_{3}^{(4)} = \frac{y′′′(3y^{′′2} − y′y′′′)}{9Ry^{′′2}} dx + \frac{y′′′^{2}}{9Ry^{′′2}} dy

μ_{4}^{(4)} = −\frac{3Ry′′y^{(4)} − y′′y^{′′′2} − 5Ry^{′′′2}}{3Ry^{′′2}(3y′′y^{(4)} − 5y^{′′′2})} dy′ −

−\frac{−20Ry^{′′′2} − 4y′′y^{′′′2} + 9Ry′′y^{(4)}}{2Ry′′(3y′′y^{(4)} − 5y^{′′′2})} dy′′ −

−\frac{y′′′(5R + y′′)}{R(3y′′y^{(4)} − 5y^{′′′2})} dy′′′

μ_{5}^{(4)} = −\frac{y′′′(3Ry′′y^{(4)} − y′′y^{′′′2} − 5Ry^{′′′2})}{3Ry^{′′2}(3y′′y^{(4)} − 5y^{′′′2})} dy′ +

+\frac{−15Ry^{′′′2} − 2y′′y^{′′′2} + 6Ry′′y^{(4)}}{Ry′′(3y′′y^{(4)} − 5y^{′′′2})} dy′′ +

+\frac{y′′′(5R + y′′)}{R(3y′′y^{(4)} − 5y^{′′′2})} dy′′′

μ_{6}^{(4)} = \frac{3y^{′′3}y′′′ − y′y′′y^{′′′2} − 5Ry′y^{′′′2} + 3Ry′y′′y^{(4)}}{9Ry^{′′3}} dx −

−\frac{−y′′y^{′′′2} − 5Ry^{′′′2} + 3Ry′′y^{(4)}}{9Ry^{′′3}} dy

where R = \sqrt{\frac{y^{′′′2}y^{′′2}}{3y^{(4)}y′′ − 5y^{′′′2}}} .

8 The final step

Now, we determine new lifted invariants by restricting the fourth order moving coframe one-forms to the curve y = y(x). This amounts to restricting dy , dy′, dy′′, dy′′′ and dy^{(4)} by their horizontal components y′dx, y′′dx, y′′′dx, y^{(4)}dx and y^{(5)}dx respectively, leading to the restricted forms

μ_{1,h}^{(4)} = −2μ_{2,h}^{(3)} + μ_{4,h}^{(3)}
μ_{2,h}^{(4)} = μ_{3,h}^{(4)} = μ_{6,h}^{(4)} = \frac{y′′′}{3R} dx

μ_{4,h}^{(4)} = \frac{9y^{′′2}y^{(5)} − 45y′′y′′′y^{(4)} + 40y^{′′′3}}{−6y′′(3y′′y^{(4)} − 5y^{′′′2})} dx
μ_{5,h}^{(4)} = −μ_{2,h}^{(3)} + 2μ_{4,h}^{(3)}

There is one non-constant linear dependency, namely μ_{4,h}^{(4)} = L μ_{3,h}^{(4)}, where

L = −\frac{sgn(y′′y′′′)(9y^{′′2}y^{(5)} − 45y′′y′′′y^{(4)} + 40y^{′′′3})}{2 (\sqrt{3y′′y^{(4)} − 5y^{′′′2}})^{3}}

is the fundamental differential invariant of the transformation group. Let J = −\frac{L}{2}. The remaining one-form

μ_{2,h}^{(4)} = H dx, H = \frac{sgn(y′′′)\sqrt{3y′′y^{(4)} − 5y^{′′′2}}}{3|y′′|} (19)

is the fundamental invariant one form. Let I = 3H. All higher order differential invariants can be found by differentiating I with respect ds; for instance, the fundamental fifth order differential invariant is D(J), where D = (D_{x})^{−1}D_{x} and

D_{x} = \frac{∂}{∂x} + y′ \frac{∂}{∂y} + y′′ \frac{∂}{∂y′} + ⋯ is the total derivative operator.

Now, the Theorem is a conclusion of Theorem 5.16 of Olver, in [6]. Therefore, as Conclusion. Every differential invariant for the group action A(2) on R^{2} is a

function of the J and its derivatives with respect to I.

References

[1] E. Cartan, La Methode du Repere Mobile, la Theorie des Groupes Continus, et les Espaces Generalises, Exposes de Geometrie No. 5, Hermman, Paris, 1935.

[2] E. Cartan, La Theorie des Groups Finis et Continus et la Geometrie differentielle Traitees par la Methode du Repere Mobile, Cahiers Scientifiques, Vol. 18, Gauthier-Villars, Paris, 1937.

[3] M. Fels and P. J. Olver, Moving coframes. I : A practical algorithm, Acta Appl. Math., Math. 51 (1998) 161-213.

[4] M. Fels and P. J. Olver, Moving coframes. II : Regularization and theoretical foundations, Acta Appl. Math., Math. 55 (1999) 127-208.

[5] S. Lie, Uber Differentialinvarianten, Math. Ann. 24 (1884), 537-578.

[6] P. J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, Cambridge University Press, 1995.

[7] A.Tresse, Sur les invariants differentiels des groupes continus de transformations, Acta Math. 18 (1894), 1-88.

Iran University of Science and Technology Department of Mathematics Narmak-16, Tehran, IRAN.