We have the polynomial $$f=x^4+x^2y^2+y^3-x^3\in\mathbb{C}[x,y].$$ Consider the ideal $I=\langle f,\frac{df}{dx},\frac{df}{dy}\rangle$. I am trying to compute the dimension of $\mathbb{C}[x,y]/I$ as a complex vector space.
I don't really know where to start, my first idea would be to calculate $\dim(\mathbb{C}[x,y])-\dim(I)$, but the first one is infinite. Any hints on how should I approach?
Macaulay2 shows that $x^2,y^2$ is a Gröbner basis for $I$, so $$\dim_{\mathbb C}\mathbb C[x,y]/I=\dim_{\mathbb C}\mathbb C[x,y]/(x^2,y^2)=4.$$