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\begin{document}

\title{Explicit computations of invariants of plane quartic curves}


\author[upb]{Andreas-Stephan Elsenhans}
\ead{elsenhan@math.uni-paderborn.de}

\address[upb]{Mathematisches Institut der Universit\"at Paderborn, Warburger Str.~100, 33098~Paderborn, Germany}


\begin{abstract}
We establish a complete set of invariants for ternary quartic forms. 
Further, we express four classical invariants in terms of these generators.
\end{abstract}

\maketitle	

\section{Introduction}

Studing rings of invariants is classical (19th century) algebra. 
For example, it was known in those days that a ternery cubic has two invariants $S$ and $T$
of degrees 4 and 6, respectively. %  that generate the ring of invariants. 
{\em Every invariant of the cubic can be expressed as a rational function
of $S$ and $T$}\/ \cite[p. 186]{Sa2}. 
Here, {\em rational}\/ should be interpreted as being a polynomial.

The ring of invariants of ternary quartics is more complicated.  
First, one can show that the degree of each invariant is divisible by 3.
In 1968, Shioda~\cite{Sh} conjectured that
the ring of invariants of ternary quartic forms is generated by 13 invariants of
degrees 3, 6, 9, 9, 12, 12, 15, 15, 18, 18, 21, 21, 27.

In 1987, Dixmier~\cite{Di} proved that invariants of degree 3, 6, 9, 12, 15, 18, 27 form
a complete system of primary invariants~\cite[Def.~2.4.6]{DK}.  
Further, he proved that at most 56 invariants suffice
to generate the entire ring of invariants.

Using the Clebsch transfer principle and invariants of binary quartics, 
we describe an efficient algorithm to compute 
invariants of the degrees given above. As an application, we show that Shioda was right.
Finally, we express four classical invariants in terms of the listed generators.

\section{Invariants, covariants, and contravariants}
The standard left action of $\Gl_n(\bbC)$ on $\bbC^n$ induces a right action on 
the polynomial ring $\bbC[X_1,\ldots,X_n]$ 
by $f^M := f(M X)$. 

Let $\bbC[X_1,\ldots,X_n]_d$ be the space of homogeneous polynomials of degree $d$. A polynomial mapping
$I \colon \bbC[X_1,\ldots,X_n]_d \rightarrow \bbC $
is called an {\em invariant} if  
$$
I(f^M) = I(f) \det(M)^e 
$$
holds for some $e \in \bbZ$ and all $M \in \Gl_n(\bbC), f \in \bbC[X_1,\ldots,X_n]_d$.
The {\em degree} of $I$ is the degree of $I$ as a polynomial in the coefficients of $f$.

A {\em covariant} $c$ is a mapping 
$c \colon \bbC[X_1,\ldots,X_n]_{d_1} \rightarrow \bbC[X_1,\ldots,X_n]_{d_2}$ with
$$
c(f^M) = c(f)^M \det(M)^e
$$
for some $e \in \bbZ$ and all $M \in \Gl_n(\bbC), f \in \bbC[X_1,\ldots,X_n]_{d_1}$.

A {\em contravariant} $C$ is a mapping
$C \colon \bbC[X_1,\ldots,X_n]_{d_1} \rightarrow \bbC[X_1,\ldots,X_n]_{d_2}$ with
$$
C(f^M) = C(f)^{(M^\top)^{-1}} \det(M)^e
$$
for some $e \in \bbZ$ and all $M \in \Gl_n(\bbC), f \in \bbC[X_1,\ldots,X_n]_{d_1}$.

We call $d_2$ the {\em order} of the convariant (resp.\ contravariant). The {\em degree} of 
a covariant (resp.\ contravariant) is the degree of its coefficients viewed as polynomials in 
the coefficients of $f$. 
\medskip

Some examples are as follows:
\begin{itemize}
\item
An invariant of a quadratic form in $n$ variables is given by the determinant of its matrix.
It is of degree $n$.
\item
More generally, the discriminant of a form of degree $d$ in $n$ variables is 
an invariant of degree $n(d-1)^{n-1}  $.
\item
A covariant of a form is given by its Hessian
$\smash{ \det\left(\frac{\partial^2 f}{\partial x_i \partial x_j}\right)_{i,j}}$.
It has degree $n$ and order $n(\deg(f) - 2)$.
\end{itemize}

\subsection*{Remarks}
\begin{itemize}
\item
Invariants are called invariants because they are invariant with respect to the action of $\Sl_n(\bbC)$.
\item
The sets of all invariants, covariants, or contravariants are rings.
\item
The ring of covariants (resp.\ contravariants) is a module over the ring of invariants.
\item
By a theorem of Hilbert, the ring of all invariants is finitely generated.
\end{itemize}
 
\section{Invariants for binary quartics}

A classical way to write down invariants is the symbolic form. 
Nowadays, this seems to be almost completely forgotten.
We refer to~\cite{Sa1} and~\cite{Sa2} for a detailed explanation
and~\cite[Chap.~VIII, Sec.~2]{We} for a more recent treatment.  
A brief summary is given in~\cite[App.~B]{Hu}.

Write a general binary quartic
in the form
 $f(x,y) := a x^4 + b x^3 y + c x^2 y^2 + d x y^3 + e y^4$.  
In degrees 2 and 3, it has the invariants 
$S_2 := 96(12ae - 3bd + c^2)$ and $T_2 := 192(72ace - 27ad^2 - 27b^2e + 9bcd - 2c^3)$.

In symbolic notation, these invariants are given by $(1 \, 2)^4$ and $(1\, 2)^2 (1\,3)^2 (2 \, 3)^2$.
Here, $(i \, j)$ is an abbreviation for the differential operator
\begin{eqnarray*}
\det
\left(
\begin{array}{cc}
\frac{\partial }{\partial x_i} & \frac{\partial }{\partial x_j} \\
\frac{\partial }{\partial y_i} & \frac{\partial }{\partial y_j}
\end{array}
\right)\, .
\end{eqnarray*}
Applying the differential operator $(12)^4$ to $f(x_1,y_1) f(x_2,y_2)$ 
gives the invariant $S_2$. Writing $T_2$ in this way, we get   
$$
(1\,2)^2 (1\,3)^2 (2\,3)^2 f(x_1,y_1) f(x_2,y_2) f(x_3,y_3)\, .
$$ 
It is a classical result that $S_2$ and $T_2$ generate the 
ring of all invariants of binary quartics~\cite[Ex.~2.1.2]{DK}.


\section{The Clebsch transfer}

 
The Clebsch transfer principle can be used to construct contravariants of ternary 
quartics from invariants of binary quartics.
For this, we denote by $(\overline{u} \, i \, j)$ the differential operator
\begin{eqnarray*}
\det \left(
\begin{array}{ccc}
u & \frac{\partial }{\partial x_i} & \frac{\partial }{\partial x_j} \\
v & \frac{\partial }{\partial y_i} & \frac{\partial }{\partial y_j} \\
w & \frac{\partial }{\partial z_i} & \frac{\partial }{\partial z_j} 
\end{array}
\right)\, .
\end{eqnarray*}
Then the contravariant $S_3(f) \in \bbC[u,v,w]$ of the ternary quartic $f(x,y,z)$ is given by 
$(\overline{u} \, 1 \, 2)^4 f(x_1,y_1,z_1) f(x_2,y_2,z_2)$.
It is of degree 2 and order 4.

Analogously, the contravariant $T_3(f)$ (degree 3, order 6) is given by 
$$(\overline{u} \, 1 \, 2)^2 (\overline{u} \, 1 \, 3)^2 
(\overline{u} \, 2 \, 3)^2 f(x_1,y_1,z_1) f(x_2,y_2,z_2) f(x_3,y_3,z_3)\, .$$


\subsection*{Interpretation}

An interpretation of these contravariants is as follows. The zero set $V(S_3(f))$ is a subset of
$(\bP^2)^\vee$. Recall that $(\bP^2)^\vee$ is the space of lines in $\bP^2$.
As a line $\ell \subset \bP^2$ can be identified with $\bP^1$, it leads to a binary quartic
$f|_\ell$. Now we get
$$
\ell \in V(S_3(f)) \Longleftrightarrow S_2(f|_\ell) = 0\, .
$$
I.e., the contravariant $S_3$ describes all lines that lead to a binary quartic 
with $S_2$ equal to zero. Analogously for $T$.

Further, the discriminant of a binary quartic is given by $S_2^3 - 6 T_2^2$. 
Thus, $S_3^3 - 6 T_3^2=0$ gives us all the lines that lead to singular binary quartics.
For a smooth quartic, this is just the dual curve.
 
\subsection*{Algorithmic aspects}

One can compute a contravariant directly from its definition. 
In the case of a ternary quartic, this performs well if one writes the differential operator
in its Horner representation.
This means, we write $(\overline{u}\, 1 \, 2)$ as
$$
 (w \frac{\partial }{\partial y_2} - v \frac{\partial }{\partial z_2})\frac{\partial }{\partial x_1}
+(u \frac{\partial }{\partial z_2} - w \frac{\partial }{\partial x_2})\frac{\partial }{\partial y_1}
+(v \frac{\partial }{\partial x_2} - u \frac{\partial }{\partial y_2})\frac{\partial }{\partial z_1}\, .
$$
Further, we multiply out as late as possible. Thus,  $T_3(f)$ becomes
$$
(\overline{u} \, 1 \, 2)^2 (\overline{u} \, 1 \, 3)^2 f(x_1,y_1,z_1) 
(\overline{u} \, 2 \, 3)^2 f(x_2,y_2,z_2) f(x_3,y_3,z_3) \, .
$$

However, there is a second method to compute these contravariants. It starts with a numerical version 
of the Clebsch transfer. For this, we denote by $\varphi_n$ the isomorphism
\begin{eqnarray*}
\varphi_n \colon \Lambda^{n-1} K^n \qquad \, \, \, \, &\rightarrow& (K^n)^\vee, \\
v_1 \wedge \ldots \wedge v_{n-1} &\mapsto& \left(x \mapsto \det(x,v_1,\ldots,v_{n-1}) \right)\,.
\end{eqnarray*}
Then we get 
$$
T_3(f)(u,v,w) = T_2(f(a_1 x + b_1 y, a_2 x + b_2 y,a_3 x + b_3 y)) \, .
$$
Here, the right hand side is the invariant $T_2$ of the binary quartic, 
given by restriction of $f$ to the line spanned by $a$ and $b$. This line is given 
via $\varphi_3^{-1}$ by $a \wedge b = \varphi_3^{-1} (u,v,w)$.

Using this, we can numerically evaluate the contravariant. Doing this at sufficiently many points
in $(\bP^3)^\vee$, one can reconstruct the contravariant by interpolation.

\smallskip

In our implementation for ternary quartics, the interpolation method is slightly faster.
A large part of the running time is used for the computation of $\varphi_3^{-1}$.
When we switch to forms in more variables, the advantage of the interpolation method 
becomes clearer. 

One can easily speed up the interpolation method by storing the used values of $\varphi_3^{-1}$ in a table.
Further, the reconstruction of the contravariant as a polynomial from its numerical values is 
done by solving a linear system of equations. As the points in $(\bP^3)^\vee$ we work with are 
a priori known, the coefficient matrix of the linear system is fixed. Thus, one could precompute 
the inverse of this matrix to speed up the interpolation step. 

\section{Invariants of ternary quartics}

A system of invariants for ternary quartics is given in~\cite{Di}, together with unpublished 
work of T.~Ohno~\cite{O}. Here, we will describe  
another system of invariants, which can easily be evaluated.
But before we can do this, we need the action of contravariants on covariants.

\subsection*{Covariant and contravariant operation}

The polynomial ring $\bbC[X_1,\ldots,X_n]$ and the ring of differential operators
$\bbC[\frac{\partial}{\partial U_1},\ldots,\frac{\partial}{\partial U_n}]$ are 
isomorphic. We fix the isomorphism $D$ given by $X_i \mapsto \frac{\partial}{\partial U_i}$.
Following~\cite[App.~B]{Hu}, we denote $D(f_1) f_2$ by $f_1 \vdash f_2$, for 
$f_1 \in \bbC[X_1,\ldots,X_n]$ and $f_2 \in \bbC[U_1,\ldots,U_n]$. 

\subsection*{Lemma}
Let $C$ be a contravariant and $c$ be a covariant.
In case the order of $c$ is larger than the order of $C$, we get a new covariant
$f \mapsto C(f) \vdash c(f)$. In case the orders coincide, this results in an invariant.
In case the order of $c$ is smaller than the order of $C$, we get the contravariant
$f \mapsto c(f) \vdash C(f)$.
 
\subsection*{A list of invariants}
Producing invariants for a form $f \in \bbC[X_1,X_2,X_3]_4$ is now somehow a random process. One multiplies known 
covariants and applies contravariants to covariants and vice versa. 
However, one can try to keep the order small such that they will be represented by
a small number of terms. Further, it might happen that some expressions degenerate to zero.
We ended up with the following: 
\begin{eqnarray*}
\begin{array}{lll}
 C_{2,0,4} := \frac{1}{96} S_3(f) &
 C_{3,0,6} := \frac{1}{192} T_3(f) &
 c_{3,6} := \det\left(\frac{\partial^2 f(x_1,x_2,x_3)}{\partial x_i \partial x_j}\right) \\
 C_{4,0,2} := f \vdash C_{3,0,6} &
 c_{4,4} := C_{2,0,4} \vdash f^2 &
 c_{5,2} := C_{4,0,2} \vdash f \\
 c_{7,4} := C_{2,0,4} \vdash (c_{4,4} \cdot f) &
 C_{7,0,2} := c_{5,2} \vdash C_{2,0,4} &
 C_{8,0,4} := c_{5,2} \vdash C_{3,0,6} \\
 c_{11,2} := C_{8,0,4} \vdash c_{3,6} &
 C_{13,0,2} := c_{5,2} \vdash C_{8,0,4} &
 c_{14,2} := C_{7,0,2} \vdash c_{7,4} \\
 c_{17,2} := C_{13,0,2} \vdash c_{4,4} &
 c_{20,2} := C_{13,0,2} \vdash c_{7,4} \\[1mm]
 I_3 := C_{2,0,4} \vdash f &
 I_6 := C_{2,0,4} \vdash c_{4,4} &
 I_{9a} := C_{4,0,2} \vdash c_{5,2} \\
 I_{9b} := C_{2,0,4} \vdash c_{7,4} & 
 I_{12a} := C_{8,0,4} \vdash c_{4,4} &
 I_{12b} := C_{7,0,2} \vdash c_{5,2} \\
 I_{15a} := C_{8,0,4} \vdash c_{7,4} &
 I_{15b} := C_{4,0,2} \vdash c_{11,2} &
 I_{18a} := C_{7,0,2} \vdash c_{11,2} \\	
 I_{18b} := C_{13,0,2} \vdash c_{5,2} &
 I_{21a} := C_{7,0,2} \vdash c_{14,2} &
 I_{21b} := C_{4,0,2} \vdash c_{17,2} \\
 I_{27} := C_{7,0,2} \vdash c_{20,2}
\end{array}
\end{eqnarray*} 


\section{Completeness of the system of invariants}

Following~\cite{Di}, a complete system of primary invariants is given by 
invariants of degree 3, 6, 9, 12, 15, 18, 27. Further, Shioda~\cite{Sh} computed the Poincare series
of the ring of invariants as
$$
\sum_{i=0}^\infty \dim(V_i) t^i =
\frac{P(t)}
{(1 - T^{27}) \cdot \prod_{i =1}^6 \left(1 - T^{3i} \right)  }
$$
with
\begin{eqnarray*}
P(t) &=& 1 + T^9 + T^{12} + T^{15} + 2 T^{18} + 3 T^{21} + 2 T^{24} + 3 T^{27}\\
&&{} + 4 T^{30} + 3 T^{33}  + 4 T^{36} +4 T^{39} + 3 T^{42} + 4 T^{45} + 3 T^{48} \\
&&{} + 2 T^{51} + 3 T^{54} + 2 T^{57} + T^{60} + T^{63} + T^{66} + T^{75}\,.
\end{eqnarray*}
Here, $V_i$ denotes the vector space of degree $i$ invariants of ternary quartics.
We get $\dim(V_i) = $
 1, 1, 2, 4, 7, 11, 19, 29, 44, 67, 98, 139, 199, 275, 375, 509, 678, 890, 1165, 
1501, 1916, 2431, 3054, 3802, 4713, 5791, 7068, 8587, 10364, 12434, 14861, 17660, 
20886, 24611 for $i = 0,3,6,9,\ldots,99$.

Further, using~\cite[Formula~3.5.1]{DK} one can read off the Poincare series that the invariants of 
degree at most $75$ generate the ring of invariants. The degree bound~\cite[Corollary~4.7.7]{DK}
for a generating system of invariants evaluates to 83 in this case. 

\subsection*{Experiment}
Using {\sc magma}~\cite{BCP}, one can compute lower bounds for the dimension of the space
of degree $d$ invariants generated by the listed invariants. 
This is done as follows:

\begin{enumerate}[(i)]
\item 
Choose a random list of quartics being longer than the expected dimension.
\item 
Compute the invariants listed above for each quartic.
\item 
Compute a spanning system (as a vector space) of all invariants of 
degree~$d$ generated by the known ones. 
For example, choose all degree $d$ monomials in a rank 13
polynomial ring with degree weights 3, 6, 9, 9, 12, 12, 15, 15, 18, 18, 21, 21, and~27. 
\item 
Compute a matrix $M$ of values of invariants 
containing one row for each quartic and one column for each vector space generator.  
\item 
Reduce this matrix $M$ modulo $p$ (e.g. $p = 101$).
\item 
Compute the rank of the matrix.
\end{enumerate}

\subsection*{Remark}
As we chose only a finite number of quartics and reduced the matrix $M$ modulo $p$, it
is not clear that we get the exact dimension of the space of invariants spanned.
However, we still get a lower bound.
 
\subsection*{Result}
As the lower bounds found coincide with the upper bound given by the coefficients of the 
Poincare series, the listed invariants in fact generate all invariants up to degree 83. 
At this place, the degree bound above shows that Shioda was right.
This was first discovered by T.~Ohno~\cite{O}.


\section{Classical invariants}
Knowing explicit generators of the ring of invariants, we can ask for a representation
of classical invariants in terms of these generators. A classical invariant is an invariant 
that has a geometric interpretation. Most of them were described in the 19th century. It is a 
somehow surprising fact that the degrees of all these invariants were known, but
explicit formulas were given only rarely.

\subsection*{Four classical invariants}
Here, we will focus on the following classical invariants: 

\begin{enumerate}[(i)]
\item
The {\em catalecticant invariant}\/ is an invariant of degree 6. 
It vanishes iff the quartic can be expressed as a sum of five 4th~powers of linear forms.
Such quartics are called {\em Clebsch quartics}.
\item
The {\em discriminant}\/ is an invariant of degree 27. It vanishes iff the quartic is singular.
\item
The {\em L\"uroth invariant}\/ is an invariant of degree 54. 
It vanishes iff the quartic can be written in the form
$l_1 l_2 l_3 l_4 + a l_1 l_2 l_3 l_5 + b l_1 l_2 l_4 l_5 + c l_1 l_3 l_4 l_5 + d l_2 l_3 l_4 l_5$.
Here, the $l_i$ are linear forms.
Such quartics are called {\em L\"uroth quartics}.
\item
The {\em Salmon invariant}\/ is an invariant of degree 60. It vanishes iff the quartic has a
flex bitangent. 
\end{enumerate} 
The L\"uroth invariant was recently constructed explicitly using several hours of
CPU time~\cite{BLRS}. 
The general approach for such an explicit construction is similar to our first experiment. 
The first step is to analyze the vector space of all degree $d$ invariants.
We describe the construction by two algorithms. 

\subsection*{Algorithm}
Given a generating set of the ring of invariants and a degree $d$, this algorithm computes 
a basis of the vector space of all degree $d$ invariants.
\begin{enumerate}[(i)]
\item
Generate all monomials of weighted degree $d$ in the given generators.
\item
Generate a sufficiently large list of randomly chosen quartics. 
\item
Compute the invariants of these quartics.
\item
Build up the matrix $M$ with one row per quartic and one column per monomial as above. 
\item
Compute the row echelon form of $M$ modulo 101. 
\item
Check that the row-space has the dimension predicted by the Poincare series. 
Otherwise, choose more quartics and redo the computation.
\item
Select all the monomials listed in step i) that correspond to the leading coefficients 
of the row echelon form  of $M$.
Return these monomials as a vector space basis of all degree $d$ invariants.
\end{enumerate}

\smallskip

Using this as a subalgorithm we can construct the invariant explicitly as follows.

\subsection*{Algorithm}
Given the degree $d$ of an invariant and its geometric interpretation, this algorithm computes an
expression of the invariant in terms of given generators. 
 
\begin{enumerate}[(i)]
\item
Compute a vector space basis $B$ of all degree $d$ invariants with the algorithm above. 
\item
Compute a list of special quartics with $\bbQ$-coefficients.
The invariant, we search for, should vanish for all these quartics.
In our cases, this means to write down a list of random Clebsch quartics, 
or quartics being singular in $[1:0:0]$, or L\"uroth quartics, 
or quartics meeting the line $z=0$ only in $[1:0:0]$.
\item
Evaluate the generators of the invariant ring for these quartics.
\item
Compute the matrix $M$ consisting of one row for each invariant in $B$ and
one column for each special quartic. 
\item
Compute the kernel of $M$. In the case it is not 1-dimensional, redo
the computation with more quartics.
\item
Compute a generator $v$ of the kernel.
\item
Return $I := \sum_{i = 1}^{\#B} v_i B_i$ as a representation of the special invariant.  
\end{enumerate}

\subsection*{Results}
The algorithm gives us the expression $3 I_6-74 I_3^2$ for the 
catalecticant invariant. 
All the other formulas are not suitable for printing as they involve
many terms and large coefficients. They are available on the author's web page
{\tt http://www.staff.uni-bayreuth.de/$\sim$bt270951/}.


As the Salmon invariant is the one of the largest degree, its reconstruction is the most complex one.
The timing results are as follows.

\begin{enumerate}[(i)]
\item
The computation of all invariants lasted 70 seconds.
\item
The computation of the row echelon form of $B$ (linear algebra over $\bbZ / 101 \bbZ$) took 10 seconds. 
\item
The computation of the kernel vector $v$  (linear algebra over $\bbQ$) was done in 58 seconds.
\end{enumerate}
The computations for the other invariants were considerably faster.
 
\subsection*{Remark}
Knowing the geometric interpretation of the catalecticant invariant, one can improve 
the experiments. We describe this in detail for the check of the completeness of the system 
of invariants, as this experiment is the largest one in terms of CPU and memory usage. 

The first step is to replace the invariant $I_6$ by the catalecticant.
Further, one chooses a certain proportion of the quartics as Clebsch quartics.   

This results in a block-structure for the matrix $M$. The upper left block consists of
invariants of Clebsch quartics not involving the catalecticant invariant. The upper right block
is zero. The lower left block consists of invariants of general quartics not involving 
the catalecticant. The lower right block contains all invariants of general quartics 
that contain the catalecticant as a factor. 

As the upper right block of $M$ is zero, a lower bound for the rank 
is given by the sum of the ranks of the upper left and the lower right blocks. 
This leads to a check of the completeness of the ring of invariants
up to degree 81 using  675 seconds of CPU time and 486 MB of memory. 
The largest inspected matrix contained approx. $52.5 \cdot 10^6$ entries in $\bbZ / 101 \bbZ$.
The computations were done with {\sc magma} 2.19 on one core of a 3 GHz Intel Core 2 X9650 processor.

\section{Comparision with other systems of invariants} 
The package {\tt Echidna}~\cite{Ko} of David Kohel can compute the Dixmier-Ohno invariants of plane quartic
curves. Similarly to the method above, these invariants can easily be related to our system. However, the 
expressions become lengthly. We use small letters to denote the Dixmier-Ohne invariants. The expressions
for the first six are
\begin{eqnarray*}
i_3 &=& \frac{1}{6} I_3 \qquad i_6  = -\frac{37}{36} I_3^2 + \frac{1}{24} I_6 \qquad
i_9 = \frac{1}{3456} I_{9a} \\
j_9 &=& \frac{1295}{1944} I_3^3 - \frac{325}{5184} I_3 I_6 + \frac{1}{31104} I_{9a} + \frac{5}{3456} I_{9b} \\
i_{12} &=& \frac{1295}{5184} I_3^4 - \frac{325}{13824} I_3^2 I_6 - \frac{11}{539136}  I_3 I_{9a} + \frac{5}{9216} I_3 I_{9b} - 
\frac{1}{359424} I_{12a} \\
j_{12} &=&+ \frac{1}{22464} I_{12b} -\frac{205}{404352}  I_3 I_{9a} + \frac{5}{269568} I_{12a} + \frac{11}{269568} I_{12b}\, . \\
\end{eqnarray*}
The formulas for the other invariants are available online on the author's web page 
{\tt http://www.staff.uni-bayreuth.de/$\sim$bt270951}.


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\bibitem{Di}
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