An approach to the Jacobian Conjecture in terms of irreducibility and square-freeness

Jędrzejewicz, Piotr; Zieliński, J.

doi:10.1007/s40879-017-0145-5

Cited by 8 publications

(5 citation statements)

References 27 publications

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“…Smale [39] in 1998 listed Jacobian conjecture as the 16th of 18 great mathematical problems for the 21th century. For Jacobian conjecture there are many positive partial results, see [12,14,15,27,31,35,38,44], etc. The investigation of Jacobian conjecture leads to a stream of valuable results concerning polynomial automorphisms, as shown in survey [3] and book [40], etc.…”

Section: Introduction and 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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Section: Introduction and Main Resultsmentioning

confidence: 99%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian

Zhao

2022

Preprint

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“…Now, we will describe some topics of an approach to the conjecture in terms of irreducibility and square-freeness. For more details we refer the reader to our survey article [14].…”

Section: Connections With the Jacobian Conjecturementioning

confidence: 99%

“…Recall two questions concerning the conditions (1.1) and (1.2) in the case of a UFD, stated in [14]. We have asked if they are equivalent under some natural assumptions (like M × = H × ), and if not, can the condition (1.1) be expressed in a form of factoriality, similarly to (1.3)?…”

Section: Introductionmentioning

confidence: 99%

On properties of square-free elements in commutative cancellative monoids

et al. 2019

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We discuss various square-free factorizations in monoids in the context of: atomicity, ascending chain condition for principal ideals, decomposition, and a greatest common divisor property. Moreover, we obtain a full characterization of submonoids of factorial monoids in which all square-free elements of a submonoid are square-free in a monoid. We also present factorial properties implying that all atoms of a submonoid are square-free in a monoid.(1.1) A(M) ⊂ S(H).

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“…The Jacobian conjecture, raised by Keller [20], has been studied by many mathematicians: a partial list of related results includes [1,2,3,4,5,6,7,8,9,10,11,12,13,16,17,18,19,21,22,25,26,27,28,29,30,31,32]. A survey is given in [14,15].…”

Section: Introductionmentioning

confidence: 99%

On the two-dimensional Jacobian conjecture: Magnus' formula revisited, I

Hurst¹,

Lee²,

Li³

et al. 2022

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Let K be an algebraically closed field of characteristic 0. When the Jacobian (∂f /∂x)(∂g/∂y) − (∂g/∂x)(∂f /∂y) is a constant for f, g ∈ K[x, y], Magnus' formula from [23] describes the relations between the homogeneous degree pieces f i 's and g i 's. We show a more general version of Magnus' formula and prove a special case of the two-dimensional Jacobian conjecture as its application.

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An approach to the Jacobian Conjecture in terms of irreducibility and square-freeness

Abstract: We present some motivations and discuss various aspects of an approach to the Jacobian Conjecture in terms of irreducible elements and square-free elements.

Cited by 8 publications

References 27 publications

The necessary and sufficient conditions for the real Jacobian conjecture

The necessary and sufficient conditions for the real Jacobian conjecture

On properties of square-free elements in commutative cancellative monoids

On the two-dimensional Jacobian conjecture: Magnus' formula revisited, I

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