Multiple root of resultant

Question

Let us suppose that we have two polynomials $F_1(x,y)$ and $F_2(x,y)$. Generally speaking, each of them defines a curve on the plane and the system of polynomial equations defined by them computes the intersection points of these curves.

If, additionally, their tangents vector are collinear at some common point $(x_1, y_1)$ then, according to my experiments with Groebner Bases, their resultant $\operatorname{Res}(F_1, \, F_2)(y)$ should have a multiple root at the point $y = y_1$.

Does this fact has simple explanation?

Answer 1

This is one of the most important steps in the proof of Bézout theorem on plane curves by elimination theory.

Without loss of generality, let me work with resultants in $x$. Assume that the point $p=(0, \,0)$ belongs to both the affine curves $C_1$ and $C_2$ of equations $F_1=0$ and $F_2=0$, respectively. Then, calling $r$ the multiplicity of $C_1$ at $p$ and $s$ the multiplicity of $C_2$ at $p$, we have $$R(x)=x^{r+s} \det \Delta(x),$$ were $\Delta(x)$ is obtained from the resultant matrix by applying suitable elementary operations on its rows and columns.

This shows that $x^{r+s}$ divides $R(x)$. Moreover, a simple analysis of the matrix $\Delta(x)$ shows that $\Delta(0)$ is invertible (namely, $x$ does not divide $\det \Delta(x)$) if and only if the tangent cones of $C_1$ and $C_2$ at $p$ have no common component. So we obtain the following

Proposition. Let $C_1$, $C_2$ be two plane curves and let $p \in C_1 \cap C_2$. Then, denoting by $\nu_p(C_1, \, C_2)$ the intersection multiplicity of $C_1$, $C_2$ at $p$, we have $$\nu_p(C_1, \, C_2) \geq \operatorname{mult}_p(C_1) \operatorname{mult}_p(C_2),$$ and equality holds if and only if the tangent cones of $C_1$, $C_2$ at $p$ have no common component.

Your situation is the case where $C_1$ and $C_2$ are both smooth at $p$, namely, $r=s=1$. Then $x$ divides $R(x)$ and, additionally, $x^2$ divides $R(x)$ if and only if $C_1$ and $C_2$ have the same tangent line at $p$.

You can find all the details in Marco Manetti's lecture notes in Algebraic Geometry (in Italian, but they are very clear), see in particular Subsection 5.3.