Let C be the plane algebraic curve defined by the polynomial P in two variables with complex coefficients. The first question under investigations is, Is there some relation between the reducibility of P and number of singularities of the the plane curve C: P(x, y)=0. The answer to this question, we use topological and algebraic properties of the plane curves. The second question is, How many irreducible components the plane curve C: P(x, y)=0 has? The answer to this question is directly related to the study of the topology of the complement of C in the complex plane by using de Rham...