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On Homogenous Minimal Involutive Varieties

Published online by Cambridge University Press:  01 February 2010

L. C. O. Almeida  and
S. C. Coutinho
L. C. O. Almeida
Affiliation:
Departamento de Ciência da Computação, Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, 21945-970 Rio de Janeiro. RJBrazil, [email protected]
S. C. Coutinho
Affiliation:
Departamento de Ciência da Computação, Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, 21945-970 Rio de Janeiro. RJBrazil, [email protected], http://www.dcc.ufrj.br/~collier

Abstract

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Ѕ(2n,k) be the variety of homogeneous polynomials of degree k in 2n variables. The authors of this paper give a computer-assisted proof that there is an analytic open set Ω of Ѕ(4,3) such that the surface F = 0 is a minimal homogeneous involutive variety of ℂ4 for all F ∈ Ω. As part of the proof, they give an explicit example of a polynomial with rational coefficients that belongs to Ω.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 301 - 315
Copyright
Copyright © London Mathematical Society 2005

References

1.Arnold, V. I., Geometrical methods in the theory of ordinary differential equations (Springer, 1983).CrossRefGoogle Scholar
2.Baum, P. and Bott, R., ‘Singularities of holomorphic foliations’, J.Differential Geom. 7 (1972) 279–342.CrossRefGoogle Scholar
3.Bernstein, I.N. and Lunts, V.‘On non-holonomic irreducible D-modules’, Invent. Math. 94 (1988) 223–243.CrossRefGoogle Scholar
4.Beauville, A., Complex algebraic surfaces, 2nd edn (Cambridge University Press, 1996).CrossRefGoogle Scholar
5.Chriss, N. and Ginzburg, V., Representation theory and complex geometry (Birhäuser, Boston, 1997).Google Scholar
6.Davenport, J.H., Siret, Y. and Tournier, E., Computer algebra(Academic Press, London, 1988).Google Scholar
7.Faugère, J. C., Gianni, P., Lazard, D. and Mora, T., ‘Efficient computation of zero- dimensional Gröbner bases by change of ordering’, J.Symbolic Comput. 16 (1993) 329–344.CrossRefGoogle Scholar
8.Greuel, G.-M., Pfister, G., and Schönemann, H., ‘Singular 2.0. A computer algebra system for polynomial computations’, Centre for Computer Algebra, University of Kaiserslautern (2001), http://www.singular.uni-kl.de.Google Scholar
9.Harris, J.D., Algebraic geometry: a first course (Springer, 1993).Google Scholar
10.Hartshorne, R., Algebraic geometry, Grad. Texts in Math. 52 (Springer, 1977).CrossRefGoogle Scholar
11.Jouanolou, J. P., Equations de Pfaff algébriques, Lect. Notes in Math. 708 (Springer, 1979).CrossRefGoogle Scholar
12.Kreuzer, M. and Robbiano, L., Computational commutative algebra 1 (Springer, Berlin/Heidelberg, 2000).CrossRefGoogle Scholar
13.Lins Neto, A., Algebraic solutions of polynomial differential equations and foliations in dimension two, Lect. Notes in Math. 1345 (1988) 192–232.Google Scholar
14.Lins Neto, A. and Soares, M. G., ‘Algebraic solutions of one dimensional foliations’, J. Differential Geom. 43 (1996) 652–673.CrossRefGoogle Scholar
15.Lunts, V., ‘Algebraic varieties preserved by generic flows’, Duke Math. J. 58 (1989) 531–554.CrossRefGoogle Scholar
16.McCune, T., ‘A generic cubic surface contains no involutive curve’, J. Algebra 235 (2001) 36–44.CrossRefGoogle Scholar
17.Shafarevich, I., Basic algebraic geometry (Springer, Heidelberg, 1977).Google Scholar
18.Soares, M. G., ‘On algebraic sets invariant by one-dimensional foliations on CP(3)’, Ann. Inst. Fourier 43 (1993) 143–162.CrossRefGoogle Scholar
19.Suwa, T., ‘Indices of holomorphic vector fields relative to invariant curves on surfaces’, Proc. Amer. Math. Soc. 123 (1995) 2989–2997.CrossRefGoogle Scholar
Supplementary material: File

JCM 8 Almeida and Coutinho Appendix A

Almeida and Coutinho Appendix A

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JCM 8 Almeida and Coutinho Appendix A readme

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