A singular point P of multiplicity r on an affine plane algebraic curve [C.sup.*] is called ordinary if and only if the r tangents to [C.sup.*] at P are distinct and nonordinary otherwise.

The criteria for the multiplicity of a point on the projective plane algebraic curve can be characterized as follows.

Q [member of] [P.sup.2](C) is a point of multiplicity r on the projective plane algebraic curve F(x, y, z) = 0 if and only if all the (r - 1)th derivatives of F(x, y, z), but not all the rth derivatives, vanish at Q.

Q [member of] [P.sup.2](C) is a singular point of the projective plane algebraic curve F(x, y, z) = 0 if and only if

Consider projective plane algebraic curve F(x, y, z) = 0 and corresponding affine plane algebraic curve f(x, y) = 0.

The complexity of Algorithm 1 is polynomial time in the degree n of the projective plane algebraic curve and is O([n.sup.5]).

We also outline the algorithm on computing the singular points of projective plane algebraic curves, and afterwards we analyze feasibility and complexity of the algorithm.

The following theorems and remarks will show the feasibility of Algorithm 1 for computing the singular points of reducible projective plane algebraic curves.

The authors [27] conclude that random perturbations of coefficients of plane algebraic curves will almost invariably destroy all singular points.

This paper provides an effective algorithm for computing the singular points of projective plane algebraic curves by homotopy continuation methods.