Irreducibility criterion for algebroid curves

Shibuta, Takafumi

doi:10.1090/S0025-5718-2012-02607-X

Irreducibility criterion for algebroid curves

Author: Takafumi Shibuta
Journal: Math. Comp. 82 (2013), 531-554
MSC (2010): Primary 14H50, 14Q05; Secondary 14H20, 13F25
DOI: https://doi.org/10.1090/S0025-5718-2012-02607-X
Published electronically: June 4, 2012
MathSciNet review: 2983035
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to give an algorithm for deciding the irreducibility of reduced algebroid curves over any algebraically closed field without using resolution of singularities. To do this, we introduce a new notion of local tropical variety which is a straightforward extension of tropism introduced by Maurer, and we prove an analogue of the the fundamental theorem of tropical geometry.

References

Shreeram S. Abhyankar, Irreducibility criterion for germs of analytic functions of two complex variables, Adv. Math. 74 (1989), no. 2, 190–257. MR 997097, DOI https://doi.org/10.1016/0001-8708%2889%2990009-1

R. Bahloul, N. Takayama, Local Gröbner fans: polyhedral and computational approach, preprint (2004), arXiv:math/0412044v2.

Robert Bieri and J. R. J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math. 347 (1984), 168–195. MR 733052

T. Bogart, A. N. Jensen, D. Speyer, B. Sturmfels, and R. R. Thomas, Computing tropical varieties, J. Symbolic Comput. 42 (2007), no. 1-2, 54–73. MR 2284285, DOI https://doi.org/10.1016/j.jsc.2006.02.004

David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. An introduction to computational algebraic geometry and commutative algebra. MR 1189133

David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811

Charles Delorme, Sous-monoïdes d’intersection complète de $N.$, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 145–154. MR 407038

Jan Draisma, A tropical approach to secant dimensions, J. Pure Appl. Algebra 212 (2008), no. 2, 349–363. MR 2357337, DOI https://doi.org/10.1016/j.jpaa.2007.05.022

Manfred Einsiedler, Mikhail Kapranov, and Douglas Lind, Non-Archimedean amoebas and tropical varieties, J. Reine Angew. Math. 601 (2006), 139–157. MR 2289207, DOI https://doi.org/10.1515/CRELLE.2006.097

Jean Fresnel and Marius van der Put, Rigid analytic geometry and its applications, Progress in Mathematics, vol. 218, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2014891

Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, Second, extended edition, Springer, Berlin, 2008. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann; With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2363237

Abramo Hefez and Marcelo Escudeiro Hernandes, Computational methods in the local theory of curves, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2001. 23$^{\rm o}$ Colóquio Brasileiro de Matemática. [23rd Brazilian Mathematics Colloquium]. MR 1849596

A. Hefez and M. E. Hernandes, Standard bases for local rings of branches and their modules of differentials, J. Symbolic Comput. 42 (2007), no. 1-2, 178–191. MR 2284291, DOI https://doi.org/10.1016/j.jsc.2006.02.008

Anders Nedergaard Jensen, Hannah Markwig, and Thomas Markwig, An algorithm for lifting points in a tropical variety, Collect. Math. 59 (2008), no. 2, 129–165. MR 2414142, DOI https://doi.org/10.1007/BF03191365

Ernst Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. 25 (1970), 748–751. MR 265353, DOI https://doi.org/10.1090/S0002-9939-1970-0265353-7

Tzee Char Kuo, A simple algorithm for deciding primes in $K[\![x,y]\!]$, Canad. J. Math. 47 (1995), no. 4, 801–816. MR 1346164, DOI https://doi.org/10.4153/CJM-1995-041-9

Joseph Maurer, Puiseux expansion for space curves, Manuscripta Math. 32 (1980), no. 1-2, 91–100. MR 592712, DOI https://doi.org/10.1007/BF01298184

Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856

Lorenzo Robbiano and Moss Sweedler, Subalgebra bases, Commutative algebra (Salvador, 1988) Lecture Notes in Math., vol. 1430, Springer, Berlin, 1990, pp. 61–87. MR 1068324, DOI https://doi.org/10.1007/BFb0085537

David Speyer and Bernd Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004), no. 3, 389–411. MR 2071813, DOI https://doi.org/10.1515/advg.2004.023

Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949

Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics, vol. 97, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002. MR 1925796

Jeremy Teitelbaum, The computational complexity of the resolution of plane curve singularities, Math. Comp. 54 (1990), no. 190, 797–837. MR 1010602, DOI https://doi.org/10.1090/S0025-5718-1990-1010602-1

N. Touda, Local Tropical Variety, preprint (2005), arXiv:math.AG/0511486.

Volker Weispfenning, Comprehensive Gröbner bases, J. Symbolic Comput. 14 (1992), no. 1, 1–29. MR 1177987, DOI https://doi.org/10.1016/0747-7171%2892%2990023-W