Affine Plane Curves

Definition: Let $K$ be a field. Two polynomials $F, G \in K[x, y]$ are said to be Equivalent if there exists a nonzero $\lambda \in K$ such that $F = \lambda G$. This forms an equivalence relation on the set of polynomials in $K[x, y]$. An Affine Plane Curve is an equivalence class of such nonconstant polynomials.

For example, $y - x^2 = 0$ is an affine plane curve, and this curve is equivalent to the curves $\lambda (y - x^2) = 0$ for every nonzero $\lambda \in K$.

We now define some characteristics of affine plane curves.

Definition: Let $K$ be a field. If $F \in K[x, y]$ is an affine plane curve and $F = \prod_{i=1}^{n} F_i^{m_i}$ where each $F_i$ is an irreducible polynomial, then each $F_i$ is called a Component of $F$ with Multiplicity $m_i$. 1) A component $F_i$ is said to be a Simple Component if $m_i = 1$. 2) A component $F_i$ is said to be a Multiple Component if $m_i \geq 2$.

For example, consider the following affine plane curve:

(1)

\begin{align} \quad (x + 2)(y^3 + 2x)^2 = 0 \end{align}

The component $x + 2$ is a simple component, while the component $y^3 + 2x$ is a multiple component.

Definition: Let $K$ be a field. The Degree of an affine plane curve is the degree of any polynomial which defines the curve. 1) A Line is an affine plane curve of degree $1$. 2) A Conic is an affine plane curve of degree $2$. 3) A Cubic is an affine plane curve of degree $3$. 4) A Quartic is an affine plane curve of degree $4$.

Examples of lines, conics, cubics, and quartics are respectively:

(2)

\begin{align} \quad x + 2y + 1 = 0 \quad , \quad x^2 - yx = 0 \quad , \quad xy^2 - 2 = 0 \quad , \quad x^4 + y^4 = 0 \end{align}