Q: Intersection multiplicity of the curves I want to find the intersection multiplicity of the curves $f(x,y)=x^5+x^4+y^2$ and $g(x,y)=x^6-x^5+y^2$ at the point $P=(0,0)$.
That`s what I have tried:
$f$ and $g$ have a common tangent, the $y=0$.
So $I(P, f\cap g) > m_P(f) \cdot m_P(g)=4$
$$f(x, 0)=x^5+x^4 \Rightarrow s=\deg f(x, 0)=5$$
$$g(x, 0)=x^6-x^5 \Rightarrow r =\deg g(x, 0)=6$$
$$s \leq r$$
So we consider $h(x, y)=g(x, y)-x f(x, y)$
$$h(x, y)=x^6-x^5+y^2-x(x^5+x^4+y^2)=x^6-x^5+y^2-x^6-x^5-xy^2 \\ \Rightarrow h(x, y)=-2x^5+y^2-xy^2$$
$\deg h(x, 0)=5<r$
So $I(P, f\cap g)=I(P,f\cap h)$
$$f(x,0)=x^5+x^4\Rightarrow \deg f(x,0)=5=s$$
$$h(x,0)=-2x^5\Rightarrow \deg h(x,0)=5=p$$
They have a common tangent, $x=0$, so they don`t intersect traverrsally.
We consider the polynomial $$h_1(x,y)=h(x,y)+2f(x,y)=3y^2-xy^2+2x^4$$
$deg h_1(x,0)=4<s,p$
So, $I(P, f\cap h)=I(P,f\cap h_1)$
$f(x,0)=x^5+x^4 \Rightarrow \deg f(x,0)=5=s$
$h_1(x,0)=2x^4\Rightarrow \deg h_1(x,0)=4=t$
They have a common tangent $x=0$,so they don`t intersect traversally.
We consider the polynomial $h_2(x,y)=2f(x,y)-xh_1(x,y)=2x^4+2y^2-3xy^2+x^2y^2$
$\deg h_2(x,0)=4<s$
So $I(P, f\cap h_1)=I(P,  h_1\cap h_2)$
$h_1(x,0)=2x^4\Rightarrow \deg h_1(x,0)=4=s$
$h_2(x,0)=2x^4\Rightarrow \deg h_2(x,0)=4=m$
They have a common tangent $x=0$, so they don`t intersect traversally.
We consider the polynomial $h_3(x,y)=h_1(x,y)-h_2(x,y)=y^2(1+2x-x^2)$
$\deg h_3(x,0)=0<s,m$
So $I(P,h_1\cap h_2)=I(P,h_2\cap h_3)$
$h_2(x,0)=2x^4\Rightarrow \deg h_2(x,0)=4=m$
$h_3(x,0)=0\Rightarrow \deg h_3(x,9)=9=n$
So $I(P,h_2\cap h_3)=I(P,h_2\cap y^2)+I(P,h_2\cap (1+2x-x^2))$
$I(P,h_2\cap y^2)=8$
$I(h_2\cap (1+2x-x^2))=0$ 
Therefore, $I(P, f \cap g)=8$.
Is it right? Do we find that $f$ and $h$ have a common tangent from $f(x,0)$ and $h(x,0)$ ?

A: See here for the definining properties of the intersection property. These allow us to compute:
$~~~I_P(x^5+x^4+y^2,x^6-x^5+y^2)\\
=I_P(x^5+x^4+y^2,(x^6-x^5+y^2)-(x^5+x^4+y^2))\\
=I_P(x^5+x^4+y^2,x^4 (x^2-2x-1))\\
=I_P(x^5+x^4+y^2,x^4)+I_P(x^5+x^4+y^2,x^2-2x-1)\\
=I_P(x^5+x^4+y^2,x^4)\\
=I_P(y^2,x^4)\\
=2 \cdot 4  \cdot I_P(y,x)\\
=8$

A: $ f(x,y) =0 $ (red curve) has no real common tangent with $ g(x,y) =0 $ (blue curve) at $ P(0,0) $. 


A: From the system of equations,
$$x^5+x^4+y^2=x^6-x^5+y^2=0,$$
you can eliminate $y$, and
$$-x^6+2x^5+x^4=0.$$
This polynomial has a quadruple root at $0$.
