New classes of polynomial maps satisfying the real Jacobian conjecture in ℝ2

Itikawa, Jackson; Llibre, Jaume

doi:10.1590/0001-3765201920170627

Cited by 4 publications

(4 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Itikawa et al [26] give two new classes of polynomial maps satisfying the real Jacobian conjecture in R 2 . A new proof of Pinchuk map which is a non-injective can be found in [2].…”

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%

See 1 more Smart Citation

The necessary and sufficient conditions for the real Jacobian conjecture

Tian

Zhao

2022

Preprint

View full text Add to dashboard Cite

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

show abstract

“…Itikawa et al [26] give two new classes of polynomial maps satisfying the real Jacobian conjecture in R 2 . A new proof of Pinchuk map which is a non-injective can be found in [2].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian

Zhao

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Itikawa et al [21] give two new classes of polynomial maps satisfying the real Jacobian conjecture in R 2 . A new proof of Pinchuk map which is a non-injective can be found in [2].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian,

Zhao

2020

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…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.…”

mentioning

confidence: 99%

Jacobian conjecture in $\mathbb R^2$

Zhang¹

2020

Preprint

View full text Add to dashboard Cite

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.

show abstract

New classes of polynomial maps satisfying the real Jacobian conjecture in ℝ2

Cited by 4 publications

References 6 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 $\mathbb R^2$

Contact Info

Product

Resources

About