On Fulton's Algorithm for Computing Intersection Multiplicities

As pointed out by Fulton in his Intersection Theory, the intersection multiplicities of two plane curves V(f) and V(g) satisfy a series of 7 properties which uniquely define I(p;f,g) at each point p∈V(f,g). Moreover, the proof of this remarkable fact is constructive, which leads to an algorithm, that we call Fulton's Algorithm. This construction, however, does not generalize to n polynomials f1, …, fn. Another practical limitation, when targeting a computer implementation, is the fact that the coordinates of the point p must be in the field of the coefficients of f1, …, fn. In this paper, we adapt Fulton's Algorithm such that it can work at any point of V(f,g), rational or not. In addition, we propose algorithmic criteria for reducing the case of n variables to the bivariate one. Experimental results are also reported.