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A fast algorithm for curve singularities

Published online by Cambridge University Press:  17 April 2009

Sheng-Ming Ma
Sheng-Ming Ma
Affiliation:
Department of Mathematics, School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, Peoples Republic of China
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We demonstrate fast algorithm to resolve local singularities of algebraic curves. The algorithm is based on the monomial transform and is independent of any other coordinate change. Two new invariants are introduced to gauge the singularities and sharply control the number of algorithmic steps. Our algorithm is applicable to both real and complex domains.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 3 , June 2004 , pp. 403 - 414
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Arnold, V., Gusein-Zade, S. and Varchenko, A., Singularities of differentiable maps I and II, Monographs in Mathematics 83 (Birkhäser, Boston, M.A., 1988).CrossRefGoogle Scholar
[2]Brieskorn, E. and Knörrer, H., Plane algebraic curves (Birkhäser Verlag, Basel, 1986).CrossRefGoogle Scholar
[3]Chudnovsky, D. and Chudnovsky, G., ‘On expansion of algebraic functions in power and Puiseux series I’, J. Complexity 2 (1986), 271–294.CrossRefGoogle Scholar
[4]Fulton, W., Algebraic curves, Mathematics Lecture Note Series (W.A. Benjamin, Inc., New York, Amsterdam, 1969).Google Scholar
[5]Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings, Lecture Notes in Mathematics 339 (Springer-Verlag, Berlin, New York, 1973).CrossRefGoogle Scholar
[6]Noether, M., ‘Ueber die algebraischen Functionen einer und zweier Variabeln’, (in German), Gött. Nachr. (1871), 267–278.Google Scholar
[7]Oka, M., ‘Geometry of plane curves via toroidal resolutions’, in Algebraic geometry and singularities, (Campillo, A., Editor), Prog. Math. 134 (Birkhäser, Basel, 1996), pp. 95–121.CrossRefGoogle Scholar
[8]Teitelbaum, J., ‘The computational complexity of the resolution of plane curve singularities’, Math. Comput. 54 190 (1990), 797–837.CrossRefGoogle Scholar
[9]Varchenko, A., ‘Newton polyhedra and estimation of oscillating integrals’, Functional Anal. Appl. 10 (1976), 175–196.CrossRefGoogle Scholar
[10]Walker, R., Algebraic curves, Princeton Mathematical Series 13 (Princeton University Press and Oxford University Press, Princeton, N.J., 1950).Google Scholar