Plane curves

A plane curve is the set of the form $\{(x,y):f(x,y)=0\}$ where $f(x,y)$ is a polynomial in two variables¹. There are many familiar examples of plane curves: for example, the circle $(x-3)^2 + (y-2)^2 = 4$ is a plane curve, as one sees by taking $f(x,y)$ to be $(x-3)^2 + (y-2)^2 - 4$.

The degree of the curve is the total degree of $f$; this is defined as the maximum of $i+j$ such that there is a monomial $a x^i y^j$ occurring in $f$ with $a \not=0$. For example, the plane curve

$$x^3 - 10 x^2 y^2 + 9 y^3 + 20 = 0$$

has degree 4 because of the monomial of largest degree in it is $-10 x^2 y^2$, which has degree $2+2=4$.