Can anyone help me solve the following exercise?
Let $p_1,p_2,p_3,p_4$ be distinct points in the projective plane $P^2$. What is the dimension of the vector space of homogeneous polynomials $f(x_0,x_1,x_2)$ of degree $n$ which pass through $p_1,p_2,p_3,p_4$? I also need to distinguish the case whether all point lie on a line or not.
What I have so far: by a standard combinatorics counting, the dimension of the vector space of homogeneous polynomials of degree $n$ is $\frac{(n+2)(n+1)}{2}$. Now, intuitively each condition $f(p_i)=0$ should decrease the dimension of the desired vector space by $1$, but how can I count how many of the $4$ conditions $f(p_i)=0$ are going to be linearly independent from each other?
Thank you in advance!