A new qualitative proof of a result on the real jacobian conjecture

Braun, Francisco; Llibre, Jaume

doi:10.1590/0001-3765201520130408

Cited by 10 publications

(6 citation statements)

References 11 publications

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“…Actually, Theorem 5 can be obtained from Theorem 4, see Section 4. Moreover, this means that Theorem 4 improves Theorem 2, and generalizes the main result of [6].…”

Section: Example 2 Consider the Polynomial Mapsupporting

confidence: 69%

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The necessary and sufficient conditions for the real Jacobian conjecture

Tian,

Zhao

2020

Preprint

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This paper is devoted to investigate the two-dimensional real Jacobian conjecture. This conjecture claims that if F = (f, g) : R 2 → R 2 is a polynomial map such that det DF (x, y) is nowhere zero and F (0, 0) = (0, 0), then F is a global injective.Firstly, we provide some new necessary and sufficient conditions such that the real Jacobian conjecture holds. By Bendixson compactification, an induced polynomial differential system can be obtained from the Hamiltonian system associated to polynomial map F . We prove that the following statements are equivalent: (A) F is a global injective; (B) the origin of induced system is a center; (C) the origin of induced system is a monodromic singular point; (D) the origin of induced system has no hyperbolic sectors; (E) induced system has a C k first integral with an isolated minimun at the origin and k ∈ N + ∪ {∞}. The above conditions (B)-(D) are local dynamical conditions.Secondly, applying the above results we present a sufficient condition for the validity of the real Jacobian conjecture. By definition a criterion function, when its limit at the origin of induced system exists, then F is a global injective. This analytical condition improves the main result of Braun et al [J. Differential Equations 260 (2016) 5250-5258]. Moreover, we use this analytical condition to give a new proof of the known algebraic sufficient condition. In this work, our all proofs are based on qualitative theory of dynamical systems.

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“…Actually, Theorem 5 can be obtained from Theorem 4, see Section 4. Moreover, this means that Theorem 4 improves Theorem 2, and generalizes the main result of [6].…”

Section: Example 2 Consider the Polynomial Mapsupporting

confidence: 69%

“…Examples 2 and 3 tell us that Theorem 5 improves Theorem 2. For s = (1, 1), Theorem 5 becomes the mentioned result of [6]. Actually, Theorem 5 can be obtained from Theorem 4, see Section 4.…”

Section: Example 2 Consider the Polynomial Mapmentioning

confidence: 91%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian,

Zhao

2020

Preprint

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“…This theorem provides a global dynamical condition such that F is a global injective. Recently, Braun and Llibre proved in [6] that if the homogeneous terms of higher degree of f and g do not have real linear factors in common and degf = degg, then F is a global injective. Later on, Braun et al [5] improved this result in the following theorem.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Notice that the sufficient condition given by Theorem 2 is not necessary [7,23], thus the paper [23] provided new sufficient conditions for the validity of the real Jacobian conjecture in which the higher homogeneous terms of the polynomials f f x + gg x and f f y + gg y have real linear factors in common of multiplicity one. Other references on this topic include the papers [8,11,20,22,31].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian

Zhao

2022

Preprint

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We focus on investigating the real Jacobian conjecture. This conjecture claims that if F = (ƒ1 , . . . , ƒn) : Rn → Rn is a polynomial map such that det DF ≠ 0, then F is a global injective. In Euclidean space Rn, the Hadamard’s theorem asserts that the polynomial map F with det DF ≠ 0 is a global injective if and only if ∥ F (x) ∥ approaches to infinite as ∥ x ∥ → ∞. This paper consists of two parts. The first part is to study the two-dimensional real Jacobian conjecture via the method of the qualitative theory of dynamical systems. We provide some necessary and sufficient conditions such that the two-dimensional real Jacobian conjecture holds. By Bendixson compactification, an induced polynomial differential system can be obtained from the Hamiltonian system associated to polynomial map F. We prove that the following statements are equivalent: (A) F is a global injective; (B) the origin of induced system is a center; (C) the origin of induced system is a monodromic singular point; (D) the origin of induced system has no hyperbolic sectors; (E) induced system has a Ck first integral with an isolated minimun at the origin and k ∈ N+∪{∞}. The above conditions (B)-(D) are local dynamical conditions. Moreover, applying the above results we present a very elementary dynamical proof of the two-dimensional Hadamard’s theorem. In the second part, we give an alternate proof of the Cima’s result on the n-dimensional real Jacobian conjecture by the n-dimensional Hadamard’s theorem 2020 Math Subject Classification: Primary 34C05. Secondary 34C08. Tertiary 14R15

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“…Along this direction, there appeared some additional sufficient conditions ensuring the real Jacobian conjecture holds. See for instance [9,10,11,12,17,25,31,44], parts of which were proceeded via tools from the theory of dynamical systems on the characterization of global centers for real planar polynomial vector fields. But, at the moment there is no a necessary and sufficient condition for which the real Jacobian conjecture holds.…”

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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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A new qualitative proof of a result on the real jacobian conjecture

Cited by 10 publications

References 11 publications

The necessary and sufficient conditions for the real Jacobian conjecture

The necessary and sufficient conditions for the real Jacobian conjecture

The necessary and sufficient conditions for the real Jacobian conjecture

Jacobian conjecture in $\mathbb R^2$

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