algebraic curve

A curve is algebraic when its defining Cartesian equation is algebraic, that is a polynomial in x and y. The order or degree of the curve is the maximum degree of each of its terms xy.
An algebraic curve with degree greater than 2 is called a higher plane curve.

An algebraic curve is called a circular algebraic curve, when the points (±1, ± i) are on the curve. In that case the highest degree of the Cartesian equation is divisible by (x² + y²). The circle is the only circular conic section.
The curve got its name from the fact that it contains the two imaginary circular points: replace x by x/w and y by y/w, and let the variable w go to zero, we obtain the circular points.

A bicircular algebraic curve passes twice through the points (±1, ± i). In this case the highest degree of the Cartesian equation is divisible by (x² + y²)².

When a curve is not algebraic, we call the curve (and its function) transcendental.

In the case a function is sufficiently sophisticated it is said to be a special function.