Abstract
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 f 1, …, f n . 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 f 1, …, f n . 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.
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Marcus, S., Maza, M.M., Vrbik, P. (2012). On Fulton’s Algorithm for Computing Intersection Multiplicities. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2012. Lecture Notes in Computer Science, vol 7442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32973-9_17
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DOI: https://doi.org/10.1007/978-3-642-32973-9_17
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