 Algebraic curve

An algebraic curve is a one-dimensional algebraic variety. On a smooth algebraic curve, the local ring at every closed point is a discrete valuation ring, which allows orders of zeros and poles to be recorded as a divisor on an algebraic curve.

An **algebraic curve** is a curve defined by a polynomial equation in two variables with coefficients in a given field, often a field of real or complex numbers. More formally, an algebraic curve can be described as the set of points (x, y) in the plane that satisfy a polynomial equation of the form: \[ F(x, y) = 0 \] where \( F(x, y) \) is a polynomial in two variables.