associated graded polynomial map singular kernel

In the ambient coordinate local ring $S=k[x,y{]}_{(x,y)}$, the associated graded ring is $k[X,Y]$: a rational function with denominator of nonzero constant term has initial form equal to its numerator initial form divided by that constant. This identifies each graded piece and respects products. Therefore orders add on products of nonzero elements of $S$. In particular for $f=y^2-x^3$, $\mathrm{in}(f)=Y^2$, and $\mathrm{in}(hf)=\mathrm{in}(h)Y^2$ for every nonzero $h\in S$.

The quotient map $S\to R$ sends ${\mathfrak{n}}^n$ onto ${\mathfrak{m}}^n$ for each $n$, so by [F1] it induces a surjection ${\mathrm{gr}}_{\mathfrak{n}}S\twoheadrightarrow {\mathrm{gr}}_{\mathfrak{m}}R$. Its degree-$n$ kernel consists of classes of $a\in {\mathfrak{n}}^n$ with $a\in (f)+{\mathfrak{n}}^{n+1}$. Write $a=hf+b$ with $b\in {\mathfrak{n}}^{n+1}$. If its degree-$n$ class is nonzero, it is exactly the initial form of $hf$, hence a multiple of $Y^2$. Conversely every homogeneous multiple of $Y^2$ is the initial form of a polynomial multiple of $f$. Thus the graded kernel is exactly $(Y^2)$, proving the displayed quotient directly. Since $f\in {\mathfrak{n}}^2$, the cotangent classes of $x,y$ form its degree-one basis, and the induced map is the claimed polynomial map.