Session S14 - Global Injectivity, Jacobian Conjecture, and Related Topics

Wednesday, July 14, 14:00 ~ 14:50 UTC-3

Some aspects of the complex Jacobian conjecture

Nguyen Thi Bich Thuy

Universidade Estadual Paulista, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The complex Jacobian conjecture: ``If a polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$ satisfies the condition $$ \det JF(p) \neq 0, \quad \forall p \in \C^n, $$ where $JF(p)$ is the Jacobian matrix of $F$ at $p$, then $F$ is an automorphism'' originally stated by Keller in 1939, is still open in the $2$-dimensional case even though the real case was solved negatively by Pinchuk [7] in 1994. In this talk, after a short survey on principal results and approaches to the conjecture, we will explicit the Newton polygon approach firstly discovered by Abhyankar [1] and developed by many mathematicians, especially by Nagata in his nice paper [2], that gave almost principal results of the conjecture in the $2$-dimensional case. Posteriorly, we present an approach to the conjecture using intersection homology [3],[4], [8]. The intersection homology of a singular variety constructed in [8] associated to a Pinchuk's counter-example was calculated [5]. Finally, we introduce a new concept: pertinent variables [6] in the study of the Jacobian conjecture. We offer also some relations between pertinent variables and Newton polygon approaches.

[1] Abhyankar, S., Expansion Techniques in Algebraic Geometry, Tata Institute of fundamental Research, Tata Institute, 1977.

[2] Nagata, M., Some remarks on the two-dimensional Jacobian conjecture, China J. Math. 17, (1989), 1--20.

[3] Nguyen, T.B.T., Valette, A. and Valette, G., On a singular variety associated to a polynomial mapping, Journal of Singularities volume 7, 190--204, 2013.

[4] Nguyen, T.B.T. and Ruas, M.A.S., On singular varieties associated to a polynomial mapping from $\mathbb{C}^n$ to $\mathbb{C}^{n-1}$, Asian Journal of Mathematics, v.22, p.1157--1172, 2018.

[5] Nguyen, T.B.T., Geometry of singularities of a Pinchuk's map, arXiv: 1710.03318v2, 2018.

[6] Nguyen, T.B.T., The $2$-dimensional Complex Jacobian Conjecture under the viewpoint of ``pertinent variables'', ArXiv:1902.05923, 2019.

[7] Pinchuk, S., A counterexample to the strong real Jacobian conjecture, Math. Zeitschrift, 217, 1--4, (1994).

[8] Valette, A. and Valette, G., Geometry of polynomial mappings at infinity via intersection homology, Ann. I. Fourier vol. 64, fascicule 5, 2147--2163, 2014.

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