Singular Point (Algebraic Curve)

A singular point of an Algebraic Curve is a point where the curve has ``nasty'' behavior such as a Cusp or a point of self-intersection (when the underlying field $K$ is taken as the Reals). More formally, a point $(a,b)$ on a curve $f(x,y)=0$ is singular if the $x$ and $y$ Partial Derivatives of $f$ are both zero at the point $(a,b)$. (If the field $K$ is not the Reals or Complex Numbers, then the Partial Derivative is computed formally using the usual rules of Calculus.)

Consider the following two examples. For the curve

$$x^3-y^2=0,$$

the Cusp at (0, 0) is a singular point. For the curve

$$x^2+y^2=-1,$$

$(0,i)$ is a nonsingular point and this curve is nonsingular.

See also Algebraic Curve, Cusp