Dimension of $\mathbb{C}[x,y]/I$ where $I$ is generated by three polynomials.

Question

I am more looking for what concepts/techniques I should understand to solve this problem rather than explicit solution (though that definitely does not hurt). It might sound strange but I'm studying for an exam that I don't know what will be tested...

Consider $\mathbb{C}[x,y]$. Consider $I = \langle x^2 + 4x + 4, xy+x+2y+2, y^3 + 3y^2 + 3y + 1\rangle$. What is the dimension of $\mathbb{C}[x,y]/I$ as a vector space over $\mathbb{C}$?

I guess I should first convince myself that $I$ is a subring. But according to a discussion here, it might not be the case, so that's already quite confusing.

Anyway, playing with the polynomials, I get

$$I = \langle (x+2)^2 , (x+2)(y+1), (y+1)^3 \rangle$$

I sense that this could be useful since it looks better now.Then I suppose the question becomes, when we divide some $f$ by $ a(x+2)^2 + b(x+2)(y+1) + c(y+1)^3$, what could be the possible remainders (where $f,a,b,c \in \mathbb{C}[x,y]$)? Because that's how I used to think about quotient ring of one variable.

Answer 1

1. $I$ is not a subring of $\mathbb{C}[x, y],$ it is an ideal. But that's the right object you want for taking a quotient of a ring, so that's OK.

2. $I=\langle (x+2)^2, (x+2)(y+1), (y+1)^3 \rangle$ definitely looks better. What would make it even better is if we were able to get rid of the constants involved. Convince yourself that there is an automorphism of $\varphi:\mathbb{C}[x, y]\rightarrow \mathbb{C}[x, y]$ that is identity on the constant polynomials and under which $(x+2)$ goes to $x$ and $(y+1)$ goes to $y$. Then instead of computing the dimension of $\mathbb{C}[x, y]/I,$ you may as well compute the dimension of $\mathbb{C}[x, y]/\varphi(I)$ (what is $\varphi(I)$?)

3. Another good trick (which works well together with 2.): Observe that $\mathbb{C}[x,y]$ is a direct sum of subspaces given by homogeneous polynomials of fixed degree $d$ (i.e. $\mathbb{C}[x, y]=\bigoplus_{d \geq 0} V_d,$ where $V_d$ has basis $x^d, x^{d-1}y, \dots, xy^{d-1}, y^d$). If you use 2., you can compute the dimension by computing $\dim_{\mathbb{C}}V_d/\varphi(I)$ degree by degree (this will be more clear after you compute $\varphi(I)$).