The Jacobian Conjecture and Injectivity Conditions

Ponnusamy, Saminathan; Starkov, Victor V.

doi:10.1007/s40840-018-0626-9

Cited by 4 publications

(5 citation statements)

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“…The set of Keller mappings for m > 3 is significantly more complicated (see [9]). A similar to Theorem A result was obtained in [6] as a sufficient condition for any n and m.…”

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confidence: 75%

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The Jacobian conjecture: structure of Keller mappings

Starkov¹

2019

Issues of Analysis

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“…The set of Keller mappings for m > 3 is significantly more complicated (see [9]). A similar to Theorem A result was obtained in [6] as a sufficient condition for any n and m.…”

supporting

confidence: 75%

“…To describe the structure of the Keller mappings, it is important to study their compositions (see, e. g., [9] and [6]). The third part of this paper is devoted to this question.…”

mentioning

confidence: 99%

The Jacobian conjecture: structure of Keller mappings

Starkov¹

2019

Issues of Analysis

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“…In addition, there are also many other partial results on both Jacobian and strong Jacobian conjectures, and on their relations with other subjects, see e.g. the survey papers by Bass et al [6] in 1982 and Essen [23] in 2000, and the recent publications [5,7,13,14,19,21,24,26,27,29,32,35,38,40,47,48,49,50,55,56]. On the history of the Jacobian and strong Jacobian conjectures we should also mention the papers by Abhyankar and Moh [3] in 1975 for two dimensional case, by Wang [51] in 1980 for the map F with degrees no more than 2, by Bass et al [6] in 1982, Yagzhev [54] in 1980 and Drużkowski [20] in 1983 via reduction of degrees of the polynomial maps, and by Hubbers [30] on cubic maps in dimension 4.…”

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confidence: 99%

Jacobian conjecture in $\mathbb R^2$

Zhang¹

2020

Preprint

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Jacobian conjecture states that ifis a polynomial map such that the Jacobian of F is a nonzero constant, then F is injective.This conjecture is still open for all n ≥ 2, and for both C n and R n . Here we provide a positive answer to the Jacobian conjecture in R 2 via the tools from the theory of dynamical systems.Let (f, g) : R 2 (C 2 ) → R 2 (C 2 ) be a polynomial map. We denote by J(f, g) the Jacobian matrix of the map (f, g), and by D(f, g) the Jacobian of (f, g),The classical Jacobian conjecture states that if the Jacobian D(f, g) = 1, then F is injective. This conjecture was first posed as a question by Keller [33] in 1939. For more information on the history of this conjecture, see e.g. the survey paper [6,23,53]. Nowadays, the Jacobian conjecture is formulated in the next form (see e.g. Bass et al [6] and Essen [23, pages XV and 82]).Associated to the Jacobian conjecture, Randall [41] in 1983 posed the so called real Jacobian conjecture.Real Jacobian conjecture. If F : R 2 → R 2 is a polynomial map such that the Jacobian DF of F does not vanish, then F is injective. This conjecture is not correct in general, as illustrated by Pinchuck [39] in 1994, which we will recall it again later on.A general formulation in C n of the real Jacobian conjecture was posed by Smale [46] in 1998 as his 16th problem on a list of 18 open mathematical problems. For distinguishing it from the Jacobian conjecture mentioned above and according to Pinchuck [39], we call it Strong Jacobian conjecture.Strong Jacobian conjecture. If F : C n → C n is a polynomial map such that the Jacobian DF of F does not vanish, then F is injective.

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“…Injectivity of such mappings G ∈ P 2 (3) follows from injectivity of Keller mappings from P 2 (100) proved in [5]. Subclasses of Keller mappings where JC is true were described for any n and m in [7]. Theorem B.…”

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confidence: 99%

“…[7] gave wider compared to theorem B subclasses of Keller mappings. Also note that in the case n = 2, m = 3 the mappings g from Theorems A and B coincide.…”

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confidence: 99%

Structure of Keller Mappings, Two-Dimensional Case

Starkov¹

2017

Issues of Analysis

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Abstract. A Keller map is a polynomial mapping f : R n → R n (or C n → C n ) with the Jacobian J f ≡ const = 0. The Jacobian conjecture was first formulated by O. N. Keller in 1939. In the modern form it supposes injectivity of a Keller map. Earlier, in the case n = 2, the author gave a complete description of Keller maps with deg f ≤ 3. This paper is devoted to the description of Keller maps for which deg f ≤ 4. Significant differences between these two cases are revealed, which, in particular, indicate the irregular structure of Keller maps even in the case of n = 2.

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The Jacobian Conjecture and Injectivity Conditions

Abstract: One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.

Cited by 4 publications

References 25 publications

The Jacobian conjecture: structure of Keller mappings

The Jacobian conjecture: structure of Keller mappings

Jacobian conjecture in $\mathbb R^2$

Structure of Keller Mappings, Two-Dimensional Case

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