Efficient algorithms for the Riemann-Roch problem and for addition in the Jacobian of a curve

Several computational problems concerning the construction of rational functions and intersecting curves over a given curve are studied. The first problem is to construct a rational function with prescribed zeros and poles over a given curve. More precisely, let C be a smooth projective curve and assume as given an affine plane model F(x,y)=0 for C, a finite set of points P_i=(X_i, Y_i) with F (X_i, Y _i)=0 and natural numbers n_i, and a finite set of points Q_i=(X_j, Y_j) with F(X_j, Y _j)=0 and natural numbers m_j. The problem is to decide whether there is a rational function which has zeros at each point P_i of order n_i , poles at each Q_j of order m_j , and no zeros or poles anywhere else on C. One would also like to construct such a rational function if one exists. An efficient algorithm for solving this problem when the given plane curve has only ordinary multiple points is given