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

Abstract

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/sub i/=(X/sub i/, Y/sub i/) with F (X/sub i/, Y/sub i/)=0 and natural numbers n/sub i/, and a finite set of points Q/sub i/=(X/sub j/, Y/sub j/) with F(X/sub j/, Y/sub j/)=0 and natural numbers m/sub j/. The problem is to decide whether there is a rational function which has zeros at each point P/sub i/ of order n/sub i/, poles at each Q/sub j/ of order m/sub 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.<>