# Intersection multiplicity and contact order of plane curves

A standard fact about intersection multiplicities of algebraic plane curves is that, for curves $$X=V(F(x,y))$$ and $$Y=V(G(x,y))$$, if $$p\in X\cap Y$$ is a non-singular point of $$X$$ and $$Y$$ then

$$I_p(X,Y)=1$$

if and only if the tangent lines of $$X$$ and $$Y$$ at $$p$$ are distinct. In other words the curves have $$0$$-th order contact at $$p$$ (if $$F_y(p)\neq 0$$), where two curves have $$k$$-th order contact at $$p$$ if they share the point $$p$$ and the $$k$$ derivatives of $$y$$ with respect to $$x$$ at $$p$$ agree.

I believe it's true that

$$I_p(X,Y)=k$$

if and only if the curves $$X$$ and $$Y$$ have $$(k-1)$$-th order contact at $$p$$, assuming $$p$$ non-singular and that the curves do not have $$k$$-th order contact. For small values of $$k$$, one can show this by locally parameterizing $$X$$ about $$p$$ using a Taylor series, and then considering the valuation of $$G$$ evaluated at this local parameterization.

Does anyone know where I can find a reference for this fact, or if this is not true?

• A hint in case someone would like to prove this on their own: WLOG assume $p=(0,0)$. Locally parameterize the two curves about $p$ using Taylor series, i.e. there exist local parameterizations $(t,\sum_{i=1}^\infty a_it^i)$ and $(t,\sum_{i=1}^\infty b_it^i)$ such that $$F(t,\sum_{i=1}^\infty a_it^i)=G(t,\sum_{i=1}^\infty b_it^i)=0.$$ Then use the fact that $I_p(F,G)=\text{val}(F(t,\sum_{i=1}^\infty b_it^i))$, to show that $a_i=b_i$ for $1\leq i\leq n$ if and only if $I_p(F,G)=n+1$. – MR_Q Jun 4 at 20:26