Algebraic curves

An algebraic curve is the set of points in the plane that satisfy a polynomial equation in two variables. For example, the equation $y = x^2$ describes a parabola, which is an algebraic curve. More generally, any equation of the form $f(x, y) = 0$, where $f$ is a polynomial, defines an algebraic curve.

Important properties

• The degree of the polynomial determines the complexity of the curve (e.g., lines are degree 1, circles and parabolas are degree 2).

• Algebraic curves can have interesting features like intersections, cusps, or loops.

• The set of solutions can be finite or infinite, depending on the equation.