lowing result.

\[keyreduction\] Every germ as specified in §\[preamble\] is equivalent to one which, up to a parameter change, has matrix representation $$\begin{pmatrix}
1 & 0 & 0\\
q(t) & t^b & 0\\
r(t) & s(t)t^b & t^c
\end{pmatrix}$$ in suitable coordinates, with $1\le b\le c$ and $q,r,s$ polynomials such that $\deg(q)<b$, $\deg(r)<c$, $\deg(s)<c-b$, and $q(0)=r(0)=s(0)=0$.

A refined version of this statement is given in Lemma \[faber\].

We will deal with $3\times 3$ matrices with entries in ${{\mathbb{C}}}[[t]]$, that is, ${{\mathbb{C}}}[[t]]$-valued points of $\operatorname{Hom}(V,W)$, for $V$, $W$ 3-dimensional complex vector spaces with chosen bases. Every such matrix $\alpha(t)$ determines a germ in ${{\mathbb{P}}}^8$. A generator $F$ of the ideal of ${{\mathscr C}}$ will be viewed as an element of ${\text{\rm Sym}}^d W^*$, for $d=\deg{{\mathscr C}}$; the composition $F\circ\alpha(t)$, a ${{\mathbb{C}}}[[t]]$-valued point of ${\text{\rm Sym}}^d V^*$, generates the ideal of ${{\mathscr C}}\circ\alpha(t)$.

We will call matrices of the form $$\lambda(t)=\begin{pmatrix}
t^a & 0 & 0\\
0 & t^b & 0\\
0 & 0 & t^c
\end{pmatrix}$$ ‘1-PS’, as they correspond to 1-parameter subgroups of ${\text{\rm PGL}}(3)$.

We will say that two matrices $\alpha(t)$, $\beta(t)$ are equivalent if the corresponding germs are equivalent in the sense of Definition \[equivgermsnew\]. The following lemma will allow us to simplify matrix expressions of germs up to equivalence. Define the degree of the zero polynomial to be $-\infty$.

\[MPI\] Let $$h_1(t) =\begin{pmatrix}
u_1 & b_1 & c_1 \\
a_2 & u_2 & c_2 \\
a_3 & b_3 & u_3
\end{pmatrix}$$ be a matrix with entries in ${{\mathbb{C}}}[[t]]$, such that $h_1(0)=I$, and let $a\le b\le c$ be integers. Then $h_1(t)$ can be written as a product $h_1(t)=h(t)\cdot j(t)$, with $$h(t)=\begin{pmatrix}
1 & 0 & 0 \\
q & 1 & 0 \\
r & s & 1 
\end{pmatrix}
\quad,\quad j(t)=\begin{pmatrix}
v_1 & e_1 & f_1 \\
d_2 & v_2 & f_2 \\
d_3 & e_3 & v_3 
\end{pmatrix}$$ where $q$, $r$, $s$ are [*polynomial