An Algorithm for Computing Torsion Differential Forms Associated with an Isolated Hypersurface Singularity

1. Aleksandrov, A.G.: On the de Rham complex of non-isolated singularities. Functional Analiz i ego Prologeniya 22, 59–60 (1988)

   MATH  Google Scholar 

2. Aleksandrov, A.G.: Logarithmic differential forms, torsion differentials and residue. Complex Var. Appl. 50, 777–802 (2005)

   MathSciNet  MATH  Google Scholar 

3. Berger, R.: Differentialmoduln eindimensionaler localer Ringe. Math. Z. 81, 326–354 (1963)

   Article  MathSciNet  Google Scholar 

4. Berger, R.: Report on the torsion of the differential module of an algebraic curve. In Algebraic Geometry and its Applications, pp. 285–303, Springer (1994)

5. Bourqui, D., Sebag, J.: On torsion Kähler differential forms. J. Pure Appl. Algebra 222, 2229–2242 (2018)

   Article  MathSciNet  Google Scholar 

6. Buchweitz, R.-O., Greuel, G.-M.: The Milnor number and deformations of complex curve singularities. Invent. Math. 58, 241–281 (1980)

   Article  MathSciNet  Google Scholar 

7. Carbonne, P.: Sur les différentielles de torsion. J. Algebra 202, 367–403 (1998)

   Article  MathSciNet  Google Scholar 

8. Greb, D., Rollenske, S.: Torsion and cotorsion in the sheaf of Kähler differentials on some mild singularities. Math. Res. Lett. 18(1), 259–1269 (2011)

   MATH  Google Scholar 

9. Greuel, G.-M.: Der Gauss–Manin–Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten. Math. Ann. 214, 2235–266 (1975)

   Article  Google Scholar 

10. Hefez, H., Hernandes, M.E.: Standard bases for local rings of branches and their modules of differentials. J. Symb. Comput. 42, 178–191 (2007)

    Article  MathSciNet  Google Scholar 

11. Hübl, R.: A note on the torsion of differential forms. Arch. Math. 54, 142–145 (1990)

    Article  MathSciNet  Google Scholar 

12. Kapur, D., Sun, D., Wang, D.: An efficient algorithm for computing a comprehensive Gröbner system of a parametric polynomial systems. J. Symb. Comput. 49, 27–44 (2013)

    Article  Google Scholar 

13. Kpognon, K.E.: Singularités des courbes planes, module des dérivations et schéma des arcs. Univ. de Rennes, Thèse de doctora (2014)

    Google Scholar 

14. Lê, D.T.: Calcul du nombre de cycles évanouissants d’une hypersueface complexe. Ann. Inst. Fourier Grenoble 23, 261–270 (1973)

    Article  MathSciNet  Google Scholar 

15. Michler, R.: Torsion of differentials of hypersurfaces with isolated singularities. J. Pure Appl. Algebra 104, 81–88 (1995)

    Article  MathSciNet  Google Scholar 

16. Michler, R.: The dual of the torsion module of differentials. Commun. Algebra 30, 5639–5650 (2002)

    Article  MathSciNet  Google Scholar 

17. Nabeshima, K.: Stability conditions of monomial bases and comprehensive Gröbner systems. In: Proc. CASC 2012, Lecture Notes in Computer Science, Vol. 7442, pp. 248–259, Springer (2012)

18. Nabeshima, K., Tajima, S.: Computing logarithmic vector fields associated with parametric semi-quasihomogeneous hypersurface isolated singularities, In: Proc of the International Symposium on Symbolic and Algebraic Computation, pp. 291–298, ACM, (2015)

19. Nabeshima, K., Tajima, S.: Efficient computation of algebraic local cohomology classes and change of ordering for zero-dimensinal standard bases. Lecture Notes Comput. Sci. 9301, 334–348 (2015)

    Article  Google Scholar 

20. Nabeshima, K., Tajima, S.: Computing Tjurina stratifications of \(\mu \)-constant deformations via parametric local cohomology systems. Appl. Algebra Eng. Comput. Comput. 27, 451–467 (2016)

    Article  MathSciNet  Google Scholar 

21. Nabeshima, K., Tajima, S.: Solving extended ideal membership problems in rings of convergent power series via Gröbner bases, Lecture Notes in Computer Sciences Vol 9582, pp. 252–267, Springer (2016)

22. Nabeshima, K., Tajima, S.: Computing \( \mu ^{\ast } \) sequences of hypersurface isolated singularities via parametric local cohomology systems. Acta Math. Vietnamica 42, 279–288 (2017)

    Article  MathSciNet  Google Scholar 

23. Nabeshima, K., Tajima, S.: Algebraic local cohomology with parameters and parametric standard bases for zero-dimensional ideals. J. Symb. Comput. 82, 91–122 (2017)

    Article  MathSciNet  Google Scholar 

24. Nabeshima, K., Tajima, S.: Computation methods of logarithmic vector fields associated with semi-weighted homogeneous isolated hypersurface singularities. Tsukuba J. Math. 42, 191–231 (2018)

    Article  MathSciNet  Google Scholar 

25. Nabeshima, K., Tajima, S.: A new method for computing the limiting tangent space of an isolated hypersurface singularity via algebraic local cohomology. Adv. Studies Pure Math. 78, 331–344 (2018)

    Article  MathSciNet  Google Scholar 

26. Nabeshima, K., Tajima, S.: Computing logarithmic vector fields and Bruce-Roberts Milnor numbers via local cohomology classes, to appear in Proc. Australian-Japanese workshop on real and complex singularities (2017)

27. Nabeshima, K., Tajima, S.: Alternative algorithms for computing generic \(\mu ^{\ast }\)-sequences and local Euler obstructions of isolated hypersurface singularities. J Algebra Appl (2019). https://doi.org/10.1142/S0219498819501561

    Article  MathSciNet  MATH  Google Scholar 

28. Nabeshima, K., Tajima, S.: Testing zero-dimensionality of varieties at a point, to appear in Mathematics in Computer Science. arXiv:1903.12365 [cs.SC]

29. Noro, M., Takeshima, T.: Risa/Asir- A computer algebra system. In: Proc. International Symposium on Symbolic and Algebraic Computation 1992, pp. 387–396, ACM, (1992) http://www.math.kobe-u.ac.jp/Asir/asir.html

30. Pol, D.: Singularités libres, formes et résidues logarithmiques, Thése de doctorat, l’Univ. d’Angers, France (2016)

31. Pohl, T.: Differential modules with maximal torsion. Arch. Math. 57, 438–445 (1991)

    Article  MathSciNet  Google Scholar 

32. Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Facut. Sci. Univ. Tokyo Sect. IA Math. 27, 265–291 (1980)

    MathSciNet  MATH  Google Scholar 

33. Sebag, J.: A remark on Berger’s conjecture, Kolchin’s theorem and arc schemes. Archiv der Math. 108, 145–150 (2017)

    Article  MathSciNet  Google Scholar 

34. Tajima, S., Nakamura, Y.: Annihilating ideals for an algebraic local cohomology class. J. Symb. Comput. 44, 435–448 (2009)

    Article  MathSciNet  Google Scholar 

35. Tajima, S., Nakamura, Y., Nabeshima, K.: Standard bases and algebraic local cohomology for zero dimensional ideals. Adv. Stud. Pure Math. 56, 341–361 (2009)

    Article  MathSciNet  Google Scholar 

36. Tajima, S.: On polar varieties, logarithmic vector fields and holonomic D-modules. RIMS Kokyuroku Bessatsu 40, 41–51 (2013)

    MathSciNet  MATH  Google Scholar 

37. Tajima, S.: Parametric local cohomology classes and Tjurina stratifications for \( \mu \)-constant deformations of quasi-homogeneous singularities. In: Several Topics on Real and Complex Singularities, pp. 189–200, World Scientific (2014)

38. Tajima, S., Nabeshima, K.: An effective method for computing Grothendieck point residue mappings, submitted

39. Teissier, B.: Cycles évanescents, section planes et conditions de Whitney. Singularités à Cargèse, Astérisque. 7–8, 285–362 (1973)

    MATH  Google Scholar 

40. Teissier, B.: Variétés polaires I. Invent. Math. 40, 267–292 (1977)

    Article  MathSciNet  Google Scholar 

41. Zariski, O.: Characterization of plane algebroid curve whose module of differentials has maximum torsion. Proc. Nat. Acad. Sci. 56, 781–786 (1966)

    Article  MathSciNet  Google Scholar 

42. Zariski, O.: Le problème des modules pour les branches planes. Hermann, Paris (1986)

    MATH  Google Scholar 

Download references