×

Effective de Rham cohomology – the general case. (English) Zbl 1422.14064

Author’s abstract: Grothendieck has proved that each class in the de Rham cohomology of a smooth complex affine variety can be represented by a differential form with polynomial coefficients. We prove a single exponential bound on the degrees of these polynomials for varieties of arbitrary dimension. More precisely, we show that the \(p\)th de Rham cohomology of a smooth affine variety of dimension \(m\) and degree \(D\) can be represented by differential forms of degree \((pD)^{\mathcal{O}(pm)}\). This result is relevant for the algorithmic computation of the cohomology, but is also motivated by questions in the theory of ordinary differential equations related to the infinitesimal Hilbert 16th problem.

MSC:

14Q20 Effectivity, complexity and computational aspects of algebraic geometry
14F40 de Rham cohomology and algebraic geometry
68W30 Symbolic computation and algebraic computation
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations

Software:

Kronecker
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Basu, S., Computing the first few Betti numbers of semi-algebraic sets in single exponential time, J. Symbolic Comput.41(10) (2006) 1125-1154. · Zbl 1126.14065
[2] Basu, S., Algorithmic semi-algebraic geometry and topology — Recent progress and open problems, in Surveys on Discrete and Computational Geometry, , Vol. 453 (American Mathematical Society, Providence, RI, 2008), pp. 139-212. · Zbl 1145.14044
[3] Basu, S., Pollack, R. and Roy, M.-F., Algorithms in Real Algebraic Geometry, , Vol. 10 (Springer, Berlin, 2003). · Zbl 1031.14028
[4] Bierstone, E., Grigoriev, D., Milman, P. and Wlodarczyk, J., Effective Hironaka resolution and its complexity, Asian J. Math.15(2) (2011) 193-228. · Zbl 1315.14022
[5] Binyamini, G., Novikov, D. and Yakovenko, S., On the number of zeros of abelian integrals, Invent. Math.181 (2010) 227-289. · Zbl 1207.34039
[6] Brownawell, W. D., Bounds for the degrees in the Nullstellensatz, Ann. of Math. (2)126(3) (1987) 577-591. · Zbl 0641.14001
[7] Bürgisser, P. and Scheiblechner, P., Differential forms in computational algebraic geometry, in ISSAC ’07: Proc. 2007 Int. Symp. Symbolic and Algebraic Computation (ACM Press, New York, USA, 2007), pp. 61-68. · Zbl 1190.14064
[8] Bürgisser, P. and Scheiblechner, P., On the complexity of counting components of algebraic varieties, J. Symbolic Comput.44(9) (2009) 1114-1136. · Zbl 1169.14041
[9] Caniglia, L., Galligo, A. and Heintz, J., Equations for the projective closure and effective Nullstellensatz, Discrete Appl. Math.33(1-3) (1991) 11-23. · Zbl 0751.14037
[10] Collins, G. E., Quantifier elimination for real closed fields by cylindrical algebraic decompostion, in Automata Theory and Formal Languages. 2nd GI Conference, Kaiserslautern, , May 20-23, 1975, ed. Brakhage, H. (Springer, Berlin, 1975), pp. 134-183.
[11] Deligne, P. and Dimca, A., Filtrations de Hodge et par l’ordre du pôle pour les hypersurfaces singulières, Ann. Sci. École Norm. Sup.23(4) (1990) 645-656. · Zbl 0743.14028
[12] Dimca, A., On the Milnor fibrations of weighted homogeneous polynomials, Compos. Math.76(1-2) (1990) 19-47. · Zbl 0726.14002
[13] Dimca, A., On the de Rham cohomology of a hypersurface complement, Amer. J. Math.113(4) (1991) 763-771. · Zbl 0743.14029
[14] Dimca, A., Singularities and Topology of Hypersurfaces, (Springer, 1992). · Zbl 0753.57001
[15] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, , Vol. 150 (Springer, New York, 1995). · Zbl 0819.13001
[16] Eisenbud, D., Huneke, C. and Vasconcelos, W., Direct methods for primary decomposition, Invent. Math.110 (1992) 207-235. · Zbl 0770.13018
[17] Giusti, M., Heintz, J., Morais, J. E. and Pardo, L. M., When polynomial equation systems can be solved fast? in Applied Algebra, Algebraic Algorithms, and Error Correcting Codes (AAECC-11), eds. Cohen, G., Giusti, M. and Mora, T., , Vol. 948 (Springer, Berlin, 1995), pp. 205-231. · Zbl 0902.12005
[18] Giusti, M., Lecerf, G. and Salvy, B., A Gröbner free alternative for polynomial system solving, J. Compl.17(1) (2001) 154-211. · Zbl 1003.12005
[19] Griffiths, P., On the periods of certain rational integrals: I, Ann. Math. (2)90(3) (1969) 460-495. · Zbl 0215.08103
[20] Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Publ. Math. Inst. Hautes Études Sci.20 (1964) 259 pp. · Zbl 0136.15901
[21] Grothendieck, A., On the de Rham cohomology of algebraic varieties, Publ. Math. Inst. Hautes Etudes Sci.39 (1966) 93-103. · Zbl 0145.17602
[22] R. Hartshorne, Local Cohomology: A Seminar Given by A. Grothendieck, Harvard University, Fall, Lecture Notes in Mathematics, Vol. 41 (Springer, Berlin, 1967).
[23] Hartshorne, R., Ample Subvarieties of Algebraic Varieties: Notes Written in Collaboration with C. Musili, , Vol. 156 (Springer, Berlin, 1970). · Zbl 0208.48901
[24] Hartshorne, R., On the de Rham cohomology of algebraic varieties, Publ. Math. Inst. Hautes Etudes Sci45 (1975) 6-99. · Zbl 0326.14004
[25] Hartshorne, R., Algebraic Geometry (Springer, New York, 1977). · Zbl 0367.14001
[26] Heintz, J., Definability and fast quantifier elimination in algebraically closed fields, Theoret. Comp. Sci.24 (1983) 239-277. · Zbl 0546.03017
[27] Jelonek, Z., On the effective Nullstellensatz, Invent. Math.162(1) (2005) 1-17. · Zbl 1087.14003
[28] Kollár, J., Sharp effective Nullstellensatz, J. Amer. Math. Soc.1(4) (1988) 963-975. · Zbl 0682.14001
[29] Kollár, J., Effective Nullstellensatz for arbitrary ideals, J. Eur. Math. Soc.1(3) (1998) 313-337. · Zbl 0986.14043
[30] Kunz, E., Kähler Differentials, (Vieweg, Wiesbaden, 1986). · Zbl 0587.13014
[31] McCleary, J., A User’s Guide to Spectral Sequences, , Vol. 12 (Publish or Perish, Wilmington, Delaware, 1985). · Zbl 0577.55001
[32] Milnor, J., On the Betti numbers of real varieties, Proc. Amer. Math. Soc.15 (1964) 275-280. · Zbl 0123.38302
[33] Mumford, D., Algebraic Geometry I: Complex Projective Varieties, , Vol. 221 (Springer, Berlin, 1976). · Zbl 0356.14002
[34] Oaku, T. and Takayama, N., An algorithm for de Rham cohomology groups of the complement of an affine variety via \(D\)-module computation, J. Pure Appl. Algebra139 (1999) 201-233. · Zbl 0960.14008
[35] Pardo, L. M., How lower and upper complexity bounds meet in elimination theory, in Applied Algebra, Algebraic Algorithms, and Error Correcting Codes (AAECC-11), eds. Cohen, G., Giusti, M. and Mora, T., , Vol. 948 (Springer, Berlin, 1995), pp. 33-69.
[36] Sabia, J. and Solernó, P., Bounds for traces in complete intersections and degrees in the Nullstellensatz, Appl. Algebra Eng. Commun. Comput.6 (1995) 353-376. · Zbl 0844.14018
[37] Scheiblechner, P., On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety, J. Compl.23(3) (2007) 359-379. · Zbl 1127.68038
[38] Scheiblechner, P., On a generalization of Stickelberger’s theorem, J. Symbolic Comput.45(12) (2010) 1459-1470. · Zbl 1211.14065
[39] Scheiblechner, P., Castelnuovo-Mumford regularity and computing the de Rham cohomology of smooth projective varieties, Found. Comput. Math.12(5) (2012) 541-571; arXiv:0905.2212v3. · Zbl 1258.14070
[40] Scheiblechner, P., Effective de Rham cohomology: The hypersurface case, in Proc. 37th Int. Symp. Symbolic and Algebraic Computation, ISSAC ’12 (ACM, New York, NY, USA, 2012), pp. 305-310; arXiv:1112.2489v1. · Zbl 1323.14015
[41] Thom, R., Sur l’homologie des variétés algébriques réelles, in Differential and Combinatorial Topology: A Symp. in Honor of Marston Morse (Princeton University Press, 1965), pp. 255-265. · Zbl 0137.42503
[42] Walther, U., Algorithmic computation of de Rham cohomology of complements of complex affine varieties, J. Symbut. Comput.29(4-5) (2000) 795-839. · Zbl 0979.14011
[43] Walther, U., Algorithmic determination of the rational cohomology of complex varieties via differential forms, in Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering, , Vol. 286 (American Mathematical Society, Providence, RI, 2001), pp. 185-206. · Zbl 1079.14507
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.