Simple and Multiple Points of Affine Plane Curves

# Simple and Multiple Points of Affine Plane Curves

Recall from the Affine Plane Curves page that we said that two polynomials $F, G \in K[x, y]$ are said to be equivalent if there exists a nonzero $\lambda \in K$ such that $F = \lambda G$, and we said that an affine plane curve is an equivalence class of a nonconstant polynomial in $K[x, y]$ with this equivalence relation.

We will now begin to classify special points of affine plane curves.

 Definition: Let $K$ be a field and let $F \in K[x, y]$ be an affine plane curve. Let $\mathbf{p} = (a, b) \in F$, that if, $\mathbf{p}$ is a point which lies on $F$. 1) $\mathbf{p}$ is a Simple Point if either $F_x(\mathbf{p}) \neq 0$ or $F_y(\mathbf{p}) \neq 0$. 2) $\mathbf{p}$ is a Multiple Point or Singular Point if both $F_x(\mathbf{p}) = 0$ and $F_y(\mathbf{p}) = 0$.

Here, the notation $F_x$ and $F_y$ are used to denote the partial derivatives of $F$ with respect to $x$ and $F$ with respect to $y$.

For example, consider the affine plane curve $F(x, y) = x^2 - yx$. The partial derivatives of $F$ are:

(1)

Observe that $F_x = 0$ whenever $y = 2x$, and $F_y = 0$ whenever $x = 0$. Therefore, every point except $(0, 0)$ is a simple point of $F$, while the point $(0, 0)$ is a singular point of $F$.

 Definition: Let $K$ be a field and let $F \in K[x, y]$ be an affine plane curve. If $\mathbf{p} = (a, b)$ is a simple point then the Tangent Line of $F$ at $\mathbf{p}$ is the line given by the equation $F_x(\mathbf{p})(x - a) + F_y(\mathbf{p})(y - b) = 0$.

For example, consider the affine plane curve $F(x, y) = y^2 - x^3 + x$. The partial derivatives of $F$ are:

(2)
The point $\mathbf{p} = (1, 0)$ is a simple point of $F$, with:
Therefore the tangent line of $F$ at $\mathbf{p}$ is: