Complex Algebraic Curves

Abstract

The equation f (x,y) = 0, where x and y are complex variables and f is a polynomial (with complex coefficients in general), defines a subset of the complex plane ℂ² (with coordinates x and y) of dimension two, since the real dimension of a 2-dimensional complex plane is equal to 4, and equating to zero the complex number f (x,y) means equating to zero both its real part and its imaginary part; that is, we have two equations in the four real variables (Re x, Imx, Rey, Imy).