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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References
Aubry, P., Lazard, D., Moreno Maza, M.: On the theories of triangular sets. J. Symb. Comp. 28(1-2), 105–124 (1999)
Berberich, E., Emeliyanenko, P., Sagraloff, M.: An elimination method for solving bivariate polynomial systems: Eliminating the usual drawbacks. CoRR, abs/1010.1386 (2010)
Bini, D., Mourrain, B.: Polynomial test suite, http://www-sop.inria.fr/saga/POL/ (accessed: April 1, 2012)
Chen, C., Moreno Maza, M.: Algorithms for computing triangular decompositions of polynomial systems. In: Proc. ISSAC 2011, pp. 83–90. ACM (2011)
Cheng, J.-S., Gao, X.-S.: Multiplicity preserving triangular set decomposition of two polynomials. CoRR, abs/1101.3603 (2011)
Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Text in Mathematics, vol. 185. Springer, New York (1998)
Dayton, B.H., Zeng, Z.: Computing the multiplicity structure in solving polynomial systems. In: Proceedings of ISSAC 2005, pp. 116–123. ACM (2005)
Fulton, W.: Introduction to intersection theory in algebraic geometry. CBMS Regional Conference Series in Mathematics, vol. 54. Conference Board of the Mathematical Sciences, Washington, DC (1984)
Fulton, W.: Algebraic curves. Advanced Book Classics. Addison-Wesley (1989)
Kalkbrener, M.: A generalized euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comp. 15, 143–167 (1993)
Kirwan, F.: Complex algebraic curves. London Mathematical Society Student Texts, vol. 23. Cambridge University Press, Cambridge (1992)
Knapp, A.W.: Cornerstones. In: Advanced algebra. Birkhäuser Boston Inc., Boston (2007), Along with a companion volume ıt Basic algebra
Labs, O.: A list of challenges for real algebraic plane curve visualization software. In: Emiris, I.Z., Sottile, F., Theobald, T. (eds.) Nonlinear Computational Geometry, pp. 137–164. Springer, New York (2010)
Lazard, D.: Solving zero-dimensional algebraic systems. J. Symb. Comp. 15, 117–132 (1992)
Lemaire, F., Moreno Maza, M., Pan, W., Xie, Y.: When does (T) equal Sat(T)? In: Proc. ISSAC 2008, pp. 207–214. ACM Press (2008)
Lemaire, F., Moreno Maza, M., Xie, Y.: The RegularChains library. In: Ilias, S. (ed.) Maple Conference 2005, pp. 355–368 (2005)
Li, Y.L., Xia, B., Zhang, Z.: Zero decomposition with multiplicity of zero-dimensional polynomial systems. CoRR, abs/1011.1634 (2010)
Shafarevich, I.R.: Basic algebraic geometry 1, 2nd edn. Springer, Berlin (1994)
Wang, D.M.: Elimination Methods. Springer (2000)
Wu, W.T.: A zero structure theorem for polynomial equations solving. MM Research Preprints 1, 2–12 (1987)
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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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