An effective approach to Picard-Vessiot theory and the Jacobian Conjecture

Paweł Bogdan,

Zbigniew Hajto,

Elżbieta Adamus

Abstrakt

In this paper we present a theorem concerning an equivalent statement of the Jacobian Conjecture in terms of Picard-Vessiot extensions. Our theorem completes the earlier work of T. Crespo and Z. Hajto which suggested an effective criterion for detecting polynomial automorphisms of affine spaces. We show a simplified criterion and give a bound on the number of wronskians determinants which we need to consider in order to check if a given polynomial mapping with non-zero constant Jacobian determinant is a polynomial automorphism. Our method is specially efficient with cubic homogeneous mappings introduced and studied in fundamental papers by H. Bass, E. Connell, D.Wright and L. Drużkowski

Słowa kluczowe: Picard-Vessiot theory, Polynomial mapping, Jacobian Conjecture
References

1] Bass H., Connell E., Wright D., The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse, Bulletin of the American Mathematical Society, 1982, 7, pp. 287–330.

[2] Bondt M. de, Homogeneous Keller maps, Ph. D. thesis, July 2007, http://webdoc.ubn.ru.nl/mono/b/bondt−m−de/homokema.pdf.

[3] Campbell L.A., A condition for a polynomial map to be invertible, Math. Annalen, 1973, 205, pp. 243–248.

[4] T. Crespo, Z. Hajto, Picard-Vessiot theory and the Jacobian problem, Israel Journal of Mathematics, 2011, 186, pp. 401–406.

[5] L. M. Dru˙zkowski, An Effective Approach to Keller’s Jacobian Conjecture, Math. Ann., 1983, 264, pp. 303–313.

[6] L. M. Dru˙zkowski, New reduction in the Jacobian conjecture, Univ. Iagell. Acta Math., 2001, 39, pp. 203–206.

[7] O.H. Keller, Ganze Cremona Transformationen, Monatsh. Math. Phys., 1939, 47, pp. 299–306.

[8] E. R. Kolchin, Picard-Vessiot theory of partial differential fields, Proceedings of the American Mathematical Society 1952, 3, pp. 596–603.

[9] S. Smale, Mathematical Problems for the Next Century, Mathematical Intelligencer, 1998, 20, pp. 7–15.

[10] D. Yan, A note on the Jacobian Conjecture, Linear Algebra and its Applications, 2011, 435, pp. 2110–2113.

Czasopismo ukazuje się w sposób ciągły on-line.
Pierwotną formą czasopisma jest wersja elektroniczna.

Wersja papierowa czasopisma dostępna na www.wuj.pl