Injectivity of non-singular planar maps with one convex component

• Autorzy: Sabatini Marco

Identyfikatory

DOI

10.4064/ba220103-28-2

• Języki publikacji: EN

Abstrakty

EN

We prove that if a non-singular planar map Λ∈C2(R2,R2) has a convex component, then it is injective. We do not assume strict convexity.

Słowa kluczowe

EN

local invertibility; global injectivity; non-strict convexity; Jacobian Conjecture

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2021
• Tom: Vol. 69, no. 2
• Strony: 107--113
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Sabatini Marco

• Dipartimento di Matematica Università di Trento I-38123 Povo (TN), Italy

• https://orcid.org/0000-0003-3479-3451

Bibliografia

• [1] H. Bass, E. H. Connell and D. Wright, The Jacobian Conjecture, reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 287-330.
• [2] J. Bernat and J. Llibre, Counterexample to Kalman and Markus-Yamabe conjectures in dimension larger than 3, Dynam. Contin. Discrete Impuls. Systems 2 (1996), 337-379.
• [3] A. Cima, A. van den Essen, A. Gasull, E. Hubbers and F. Mañosas, A polynomial counterexample to the Markus-Yamabe conjecture, Adv. Math. 131 (1997), 453-457.
• [4] M. de Bondt and A. van den Essen, Recent progress on the Jacobian conjecture, Ann. Polon. Math. 87 (2005), 1-11.
• [5] F. Braun and B. Oréfice-Okamoto, On polynomial submersions of degree 4 and the real Jacobian conjecture in R2, J. Math. Anal. Appl. 443 (2016), 688-706.
• [6] F. Braun, J. Giné and J. Llibre, A sufficient condition in order that the real Jacobian conjecture in R2 holds, J. Differential Equations 260 (2016), 5250-5258.
• [7] F. Braun and J. R. dos Santos Filho, The real Jacobian conjecture on R2 is true when one of the components has degree 3, Discrete Contin. Dynam. Systems 26 (2010), 75-87.
• [8] A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Progr. Math. 190, Birkhäuser, Basel, 2000.
• [9] R. Fessler, A proof of the two-dimensional Markus-Yamabe stability conjecture and a generalization, Ann. Polon. Math. 62 (1995), 45-74.
• [10] A. A. Glutsyuk, Complete solution of the Jacobian problem for planar vector fields, Uspekhi Mat. Nauk 49 (1994), no. 3, 179-180 (in Russian); English transl.: Russian Math. Surveys 49 (1994), no. 3, 185-186.
• [11] C. Gutierrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 627-671.
• [12] C. Gutierrez, X. Jarque, J. Llibre and M. A. Teixeira, Global injectivity of C1 maps of the real plane, inseparable leaves and the Palais-Smale condition, Canad. Math. Bull. 50 (2007), 377-389.
• [13] J. M. Hadamard, Sur les correspondances ponctuelles, in: Oeuvres, Éditions du Centre Nationale de la Recherche Scientifique, Paris, 1968, 383-384.
• [14] O. H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), 299-306.
• [15] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. J. 12 (1960), 305-317.
• [16] C. Olech, On the global stability of an autonomous system on the plane, Contributions to Differential Equations 1 (1963), 389-400.
• [17] S. Pinchuk, A counterexample to the strong real Jacobian conjecture, Math. Z. 217 (1994), 1-4.
• [18] M. Sabatini, An extension to Hadamard global inverse function theorem in the plane, Nonlinear Anal. 34 (1998), 829-838.
• [19] M. Sabatini, Commutativity of flows and injectivity of nonsingular mappings, in: Polynomial Automorphisms and Related Topics (Kraków, 1999), Ann. Polon. Math. 76 (2001), 159-168.
• [20] S. Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), 7-15.
• [21] N. Van Chau and C. Gutierrez, Properness and the Jacobian conjecture in R2 , Vietnam J. Math. 31 (2003), 421-427.
• [22] X. Zhang, Jacobian conjecture in R 2, arXiv:2011.12701 (2020).
• [23] X. Zhang, Jacobian conjecture in R2, arXiv:2011.12701v2 (2021).
• [24] A. V. Yagzhev, Keller’s problem, Siberian Math. J. 21 (1980), 747-754.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).