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Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows: $$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \dim_{\mathbb{C}} \mathbb{C}[[x,y]]\bigg/\left(f, \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)^s\right)$$ I've noticed that in various examples (like the $A_{n-1}$-singularities $f = y^2 - x^n$, or more generally, the hypercuspidal singularities $f = y^a - x^b$) that $e_f$ is equal to the intersection multiplicity of $f$ and a ``generic polar'' of $f$. More precisely, I've observed that the following equality holds in many examples: $$e_f = \dim_{\mathbb{C}} \mathbb{C}[[x,y]]\bigg/\left(f, \alpha \cdot \frac{ \partial f}{\partial x} - \beta \cdot \frac{\partial f}{ \partial y}\right) \qquad (*)$$ where $[\alpha : \beta] \in \mathbb{P}_{\mathbb{C}}^1$ is generic. My question: is it well-known (or obvious) whether the equality $(*)$ holds for an arbitrary isolated plane curve singularity germ $f$?

(Note: the colength of the polar (the quantity on the right of $(*)$) is of interest because it is related to Teissier's notion of ``polar invariant.'')

What I have so far: Note that the polar is itself a complete intersection, so the colength of the polar in the ring $\mathbb{C}[[x,y]]/(f)$ is equal to the Hilbert-Samuel multiplicity of the polar. Since the Hilbert-Samuel multiplicity of an ideal is equal to that of any reduction, it suffices to show that the ideal $I = \left(\alpha \cdot \frac{\partial f}{\partial x} - \beta \cdot \frac{\partial f}{\partial y}\right)$ generated by the polar is a reduction of the ideal $J = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$ (here, we are regarding $I,J$ as ideals of the ring $\mathbb{C}[[x,y]]/(f)$). I.e., it suffices to show that for some $s \geq 0$, the obvious inclusion $$I(J^s) \subset J^{s+1}$$ of ideals of $\mathbb{C}[[x,y]]/(f)$ is in fact an equality. Now restrict to the case where $f \in \mathbb{C}[x,y]$ is homogeneous of degree $d$. Then it is not hard to check by counting generators that $IJ^{d-1} = (x,y)^{d(d-1)} \supset J^d$ (as ideals of $\mathbb{C}[[x,y]]/(f)$). I do not currently know how to generalize this argument to handle germs $f$ that are not homogeneous.

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First, the quantity you defined is the Hilbert-Samuel multiplicity of the ideal $J= (f_x,f_y)$ in $R=\mathbb C[[x,y]]/(f)$. The multiplicity of $f$ usually refers to the multiplicity of the maximal ideal $m$ of $R$.

As you noted, it is enough to show that a generic combination of the generators of $J$ is a reduction. This is true much more generally, and basically the point is to consider the fiber cone $S= F(I) = k\oplus \frac {J}{mJ}\oplus \frac {J^2}{mJ^2}\oplus... $. This algebra has dimension $1$, and is generated in degree one. Then, for a generic linear form $l \in S_1$, $S/lS$ is $0$-dimensional, so $lS_n=S_{n+1}$ for $n\gg 0$. But since $l \in S_1= J/mJ$, this says exactly that $J^nc = J^{n+1}$ where $c$ is a lift of $l$.

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