The Intersection Multiplicity of Intersection Points over Algebraic Curves

In analytic geometry, Bézout’s theorem stated the number of intersection points of two algebraic curves and Fulton introduced the intersection multiplicity of two curves at some point in local case. It is meaningful to give the exact expression of the intersection multiplicity of two curves at some point. In this paper, we mainly express the intersection multiplicity of two curves at some point in ${\mathbb{R}}^2$ and ${\mathbb{A}}_K^2$ under fold point, where $\text{char}\left(K\right)=0$ . First, we give a sufficient and necessary condition for the coincidence of the intersection multiplicity of two curves at some point and the smallest degree of the terms of these two curves in ${\mathbb{R}}^2$ . Furthermore, we show that two different definitions of intersection multiplicity of two curves at a point in ${\mathbb{A}}_K^2$ are equivalent and then give the exact expression of the intersection multiplicity of two curves at some point in ${\mathbb{A}}_K^2$ under fold point.