# Affine Plane Curves – The Math

 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:

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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:

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