compute the maximum power of a polynomial that divides other

Question

Let F be a polynomial function in 3 variables of degree $n$ passing trough 5 points with given multiplicities. It is well know that there exists a unique quadratic polynomial passing throug any given five points, call it $F_2$. Using Bezout Theorem it is possible to see $F_2$ is a factor of $F$. I would like to know if it is possible to determine the maximum power of $F_2$ that divides $F$, in terms of $n$ and the multiplicity of the points. It seems to me that this is just one more application of the Bezout's Theorem but I don't know how to move further. Any reference is welcome.

Answer 1

Let $a,b,c,d,e$ be the multiplicities of the points, if the points are in general position then the conic $F_2$ must be non-singular, and passes through each point exactly once. Now, you know that $F = F_2 G$ where $G$ is a polynomial of degree $n-2$. G passes trough the five given points with multiplicity $a-1,b-1,c-1,d-1,e-1$. If $F_2$ does not divide $G$, the maximum power is 1, otherwise, using the same argument (Bezout's Theorem) you see that $F_2$ divides $G$. So inductively you want to find the first $k$ such that $2(n-2k) \geq (a-k) + (b-k) + (c-k) + (d-k) + (e-k)$. Solving the inequality you have the wanted number.