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)

begin{align} quad F_x(x, y) = 2x – y quad , quad F_y(x, y) = x end{align}

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)

begin{align} quad F_x(x, y) = -3x^2 + 1 quad , quad F_y(x, y) = 2y end{align}

The point $mathbf{p} = (1, 0)$ is a simple point of $F$, with:

(3)

begin{align} quad F_x(mathbf{p}) = -2 quad , quad F_y(mathbf{p}) = 0 end{align}

Therefore the tangent line of $F$ at $mathbf{p}$ is:

(4)

begin{align} quad -2(x – 1) + 0(y – 0) = 0 \ end{align}

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