algebraic curve

An algebraic curve is a curve defined by an algebraic equation, meaning the coordinates of its points are related by a polynomial equation. This is in contrast to transcendental curves, which involve transcendental functions in their equations. Algebraic geometry, which emerged after 1850, studies the geometric properties of solutions to polynomial equations, including those in dimensions beyond three.123

An algebraic curve C is represented by an equation f(x, y) = 0, where f is a polynomial. These curves can be classified by their genus, a non-negative integer calculated from the polynomial. For example, the parabola y = x² is an algebraic curve. Algebraic curves are fundamental objects in mathematics, and their study has led to significant advancements in various fields, including number theory.24