 C}}$ are tangent to the line $y=0$, leaving to the reader the necessary adjustments in the presence of such branches. We write the generator $F$ for the ideal of ${{\mathscr C}}$ as a product of formal branches $F =\prod_{i=1}^m (z-f_i(y))$. We will focus on the formal branches that are tangent to the line $z=0$, which may be written explicitly as $$z=f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1<\lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$.

Now we begin the proof of Proposition \[standardform\].

Reduction to $q\ne 0$
---------------------

The first remark is that, under the assumptions that $q$, $r$, and $s$ do not all vanish, we may in fact assume that $q(t)$ is not identically zero.

\[qnot0\] If $\alpha(t)$ is as in $(\dagger)$, and $q= 0$, then $\lim_{t\to 0}\mathcal
C\circ\alpha(t)$ is a rank-2 limit.

(Sketch.) Assume $q=0$, and study the action of $\alpha(t)$ on individual monomials $x^Ay^Bz^C$ in an equation for ${{\mathscr C}}$: $$m_{ABC}:=x^A y^B (r(t) x+s(t) t^b y+t^c z)^C t^{bB}\quad.$$ There are various possibilities for the vanishing of $r$ and $s$, but the dominant terms in $m_{ABC}$ are always kernel stars, which are rank-2 limits by Lemma \[rank2lemma\].

Reduction to $b<c$
------------------

By Lemma \[qnot0\] and the last part of Lemma \[faber\] we may replace $\alpha(t)$ with an equivalent germ $$\begin{pmatrix}
1 & 0 & 0     \\
t^a & t^b & 0 \\
r(t) & s(t)t^b & t^c\end{pmatrix}$$ with $a<b\le c$, and $r(t)$, $s(t)$ polynomials of degree $<c$, $<(c-b)$ respectively and vanishing at $t=0$.

Next, we have to show that if $\lim_{t\to 0}{{\mathscr C}}\circ\alpha(t)$ is not a rank-2 limit then $b<c$ and $r(t)$, $s(t)$ are as stated in Proposition \[standardform\].

\[b=c\] Let $\alpha(t)$ be as above. If $b=c$, then $\lim_{t\to 0} 
{{\mathscr C}}\circ\alpha(t)$ is a rank-2 limit.

Decompose $F(1:y:z)$ in ${{\mathbb{C}}}[[y,z]]$: write $F(1:y:z)=G(y,z)\cdot H(y,z)$, where $G(y,z)$ collects the branches that are [*not*]{} tangent 