I am working on the following task from Ideals, Varieties, and Algorithms by Cox, Little, Shea:
I already completed parts a and b of the task. But I struggle finishing part c: Using the hint I take $g\in I(V)$. So $g\in K[x,y]$ and $g(a_s,b_s)=0$ $\forall 1\leq s \leq n$. So I thought because of the unambiguity of $h$ that $g$ has to be some kind of $y-h(x)$. But I am not quite sure if this is true. Especially, since $I(V)$ contains the product of $(x-a_i)$ in its generators. I'm happy about any hint!
PS: Since I am new here I would appreciate feedback on the formulation of my entry!
EDIT: Since $g(a_s,b_s)=0$ and $h$ is the unique polynomial it follows that $g(x,y)=p(x,y)*(y-h(x))$ with $p$ polynomial. Then $g(a_s,b_s)=p(a_s,b_s)*(b_s-h(a_s))=p(a_s,b_s)*0=0$. But I do not understand why $p$ has to be some kind of $(x-a_i)$...