Post date: Mar 16, 2012 1:3:50 PM
Remarkably, and as pointed out by Fulton in his Intersection Theory, the intersection multiplicities of the 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 (generating a zero-dimensional of k[x1, . . . , xn], for an arbitrary field k) for n > 2. Another practical limitation, when targeting a computer implementation, is the fact that the coordinates of the point p must be in the base field k. Approaches based on standard or Gr¨obner bases suffer from the same limitation.
In this work, we adapt Fulton’s Algorithm such that it can work at any point of V (f, g), rational or not. In addition, and under genericity assumptions, we add an 8-th property to the 7 properties of Fulton, which ensures that these 8 properties uniquely and constructively define I(p; f1, . . . , fn) at any p ∈ V (f1, . . . , fn). The implementation of this 8-th property has lead us to a new approach for computing the tangent cones that do not involve standard or Gr¨obner bases. In fact, all our algorithms simply rely on the theory of regular chains and are implemented in the RegularChains library in Maple.
Joint work with:Steffen Marcus, Paul Vrbik