The Real Jacobian Conjecture for polynomials of degree 3

Gwoździewicz, Janusz

doi:10.4064/ap76-1-12

Cited by 12 publications

(9 citation statements)

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“…In [9,10], the authors were based on the structure of polynomial maps to give sufficient conditions. Gwoździewicz in [18] obtained that the real Jacobian conjecture holds if the degrees of f and g are less than or equal to 3. Braun et al [4,8] generalized this result by showing that the conjecture is true if the degree of f is at most 4, independently of the degree of g. In the above mentioned papers, the main technique relates algebra, analysis and geometry.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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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Section: Introduction and Main Resultsmentioning

confidence: 99%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian,

Zhao

2020

Preprint

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“…In this section we present results that allow to prove smoothly Theorem 1. Most of them are taken from [3,4,5]. Only Lemmas 4, 5 and 8 are new.…”

Section: Lemmasmentioning

confidence: 99%

Real Jacobian mates

Gwoździewicz

2016

Ann. Polon. Math.

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“…The dimension 2 results were later improved to cover polynomial maps with components of degree at most 3 [13], and then to polynomial maps with one component of degree at most 3 [4].…”

Section: Consider the Extension Of G By Fresh Variables Tomentioning

confidence: 99%

Reduction theorems for the Strong Real Jacobian Conjecture

Campbell¹

2014

Ann. Polon. Math.

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Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic and geometric properties of the maps involved. That permits the separate formulation and reduction, though not so far the solution, of the SRJC for classes of nonsingular polynomial endomorphisms of real nspace that exclude the Pinchuk counterexamples to the SRJC, for instance those that induce rational function field extensions of a given fixed odd degree.

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The Real Jacobian Conjecture for polynomials of degree 3

Abstract: We show that every local polynomial diffeomorphism (f, g) of the real plane such that deg f ≤ 3, deg g ≤ 3 is a global diffeomorphism.

Cited by 12 publications

References 5 publications

The necessary and sufficient conditions for the real Jacobian conjecture

The necessary and sufficient conditions for the real Jacobian conjecture

Real Jacobian mates

Reduction theorems for the Strong Real Jacobian Conjecture

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