The intersection multiplicity of intersection points over algebraic curves (Q6098337)

It is well known, that 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. In this paper, authors deal with an important problem, namely, they express the intersection multiplicity of two curves at some point in \(\mathbb{R}^2\) and \(\mathbb{A}_{K}^2\) under fold point, where \(K\) is a field of characteristic zero. For this purpose, authors 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, it is shown that two different definitions of intersection multiplicity of two curves at a point in \(\mathbb{A}_{K}^2\) are equivalent and then provide the exact expression of the intersection multiplicity of two curves at some point in \(\mathbb{A}_{K}^2\) under fold point.