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

• Autorzy: Bogdan P. , Hajto Z. , Adamus E.

Identyfikatory

DOI

10.4467/20838476SI.17.004.8150

• Języki publikacji: EN

Abstrakty

EN

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

EN

Picard-Vessiot theory; Polynomial mapping; Jacobian Conjecture

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Schedae Informaticae
• Rocznik: 2017
• Tom: Vol. 26
• Strony: 49--59
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Bogdan P.

• Faculty of Mathematics and Computer Science, Jagiellonian University Lojasiewicza 6, 30-348 Krak´ow

autor

Hajto Z.

• Faculty of Mathematics and Computer Science, Jagiellonian University Lojasiewicza 6, 30-348 Krak´ow

autor

Adamus E.

• Faculty of Applied Mathematics, AGH University of Science and Technology al. Mickiewicza 30, 30-059 Krak´ow, Poland

Bibliografia

• [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. Drużkowski, An Effective Approach to Keller’s Jacobian Conjecture, Math. Ann., 1983, 264, pp. 303–313.
• [6] L. M. Drużkowski, 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.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).