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 degree-$n$ kernel of $\mathrm{gr}S\to \mathrm{gr}(S\mathrm{/}(f))$ 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 $(Y^2)$ and the surjective polynomial map of the cotangent-basis lemma has the stated quotient.