Branch Curves of Multiple Planes and Continuous Systems of Plane Algebraic Curves

Let z be a k-valued algebraic function of two complex variables x and y, defined by an irreducible algebraic equation, 1$$F(x,y,z){\rm{ }} = {\rm{ }}0$$. The branch curvef, 2$$f(x,y){\rm{ }} = {\rm{ }}0$$, of the function z is found by eliminating z between F = 0 and $$\partial F/\partial z = 0$$ and by neglecting in the resultant certain factors which correspond to multiple curves of the surface F = 0 (apparent branch curves) The definition of f may be rendered exact by assuming that: (a) the polynomial f contains no multiple factors; (b) the curve f is the locus of the effective branch points (x₁, y), (x₂, y),…,(x_n, y) of the function z = z(x,y), for y fixed, and of the lines y = c = const. such that y = c is an effective branch point of z if x is fixed and generic. It may be necessary to include the line at infinity of the projective plane (x,y) in the branch curve. However, we may always choose the coördinates x and y in such a manner that the line at infinity does not belong to the branch curve.