The Multiplicity of an Affine Plane Curve at a Point

Definition: Let $K$ be a field and let $F \in K[x, y]$ be an affine plane curve. Let $F = F_m + F_{m+1} + ... + F_n$ where each $F_i$ is a form of degree $i$. The Multiplicity of $F$ at $\mathbf{p} = (0, 0)$ is defined to be $m_{\mathbf{p}}(F) = m$.

In other words, the multiplicity of an affine plane curve at $\mathbf{p} = (0, 0)$ is defined to be the smallest degree of all forms which comprise $F$.

For example, consider the curve $F(x, y) = y^2 - x^2y + xy$. We have that $F_2(x, y) = y^2 + xy$ and $F_3(x, y) = -x^2y$ so that $F = F_2 + F_3$. Therefore the multiplicity of $F$ at $\mathbf{p} = (0, 0)$ is:

(1)

\begin{align} \quad m_{(0, 0)}(y^2 - x^2y + xy) = 2 \end{align}

Theorem 1: Let $K$ be a field and let $F \in K[x, y]$ be an affine plane curve. Then $\mathbf{p} = (0, 0)$ is a simple point of $F$ if and only if $m_{\mathbf{p}}(F) = 1$.

• Proof: $\Rightarrow$ Suppose that $\mathbf{p} = (0, 0)$ is a simple point of $F$. Then either $F_x(0, 0) \neq 0$ or $F_y(0, 0) \neq 0$. Suppose that $m_{\mathbf{p}}(F) = n > 1$. Then $F$ contains terms of the form $ax^iy^j$ where $a \in K$ and $i + j \geq n$. So $F_x$ and $F_y$ contain terms of the form $aix^{i-1}x^j$ and $ajx^iy^{j-1}$ respectively. Evaluating each term at $\mathbf{p} = (0, 0)$ gives us that $F_x(0, 0) = 0$ and $F_y(0, 0) = 0$ which is a contradiction since $i-1 + j \neq 0$. So we must have that:

(2)

\begin{align} \quad m_{\mathbf{p}}(F) = 1 \end{align}

• $\Leftarrow$ Suppose that $m_{\mathbf{p}}(F) = 1$. Then $F$ contains a term of the form $ax^i + by^j$ where $a, b \in K$, $a, b$ not both zero, $i, j = 0, 1$ and $i, j$ not both zero. Therefore $F_x$ or $F_y$ contains a term of the form $a$ or $b$, so either $F_x(0, 0) \neq 0$ or $F_y(0, 0) \neq 0$, so $\mathbf{p} = (0, 0)$ is a simple point of $F$. $\blacksquare$

Corollary 2: Let $K$ be a field and let $F \in K[x, y]$ be an affine plane curve. If $\mathbf{p} = (0, 0)$ is a simple point then the tangent line of $F$ at $\mathbf{p} = (0, 0)$ is the form $F_1$ of $F$.

For example, consider the affine plane curve $F(x, y) = y^2 - x^3 + x$. Then the multiplicity of $F$ at $\mathbf{p} = (0, 0)$ is $m_{\mathbf{p}}(F) = 1$ so by Theorem 1 $\mathbf{p} = (0, 0)$ is a simple point of $F$ and by corollary 2 we have that the tangent line to $F$ at $\mathbf{p} = (0, 0)$ is $x = 0$.

Definition: Let $K$ be a field and let $F \in K[x, y]$ be an affine plane curve and let $\mathbf{p} = (0, 0)$. 1) $\mathbf{p}$ is a Simple Point of $F$ if $m_{\mathbf{p}}(F) = 1$. 1) $\mathbf{p}$ is a Double Point of $F$ if $m_{\mathbf{p}}(F) = 2$. 1) $\mathbf{p}$ is a Triple Point of $F$ if $m_{\mathbf{p}}(F) = 3$.