On the shape of possible counterexamples to the Jacobian Conjecture

Valqui, Christian; Guccione, Jorge A.; Guccione, Juan J.

doi:10.1016/j.jalgebra.2016.08.039

Cited by 14 publications

(47 citation statements)

References 12 publications

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“…, X n , and hence X i ∈ I F for every i, as needed. Therefore, in other words, if there exists a Keller map F such that I F is strictly contained in the ideal generated by the indeterminates, then F would be a counterexample to JC (for a detailed study about the shape of possible counterexamples in the two-dimensional case, see [25]). This is particularly intriguing if k is not algebraically closed, since in this case I F might eventually admit an associated maximal ideal M ⊃ I F such that V (M ) is empty; for instance, it is not known whether the base ideal of a planar Keller map F over the real number field R can be decomposed as…”

Section: Conjecture (C) If F Is Keller Then Imentioning

confidence: 99%

An ideal-theoretic approach to Keller maps

Miranda-Neto¹

2019

Proceedings of the Edinburgh Mathematical Society

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A self-map F of an affine space ${\bf A}_k^n $ over a field k is said to be a Keller map if F is given by polynomials F1, …, Fn ∈ k[X1, …, Xn] whose Jacobian determinant lies in $k\setminus \{0\}$. We consider char(k) = 0 and assume, as we may, that the Fis vanish at the origin. In this note, we prove that if F is Keller then its base ideal IF = 〈F1, …, Fn〉 is radical (a finite intersection of maximal ideals in this case). We then conjecture that IF = 〈X1, …, Xn〉, which we show to be equivalent to the classical Jacobian Conjecture. In addition, among other remarks, we observe that every complex Keller map admits a well-defined multidimensional global residue function.

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Section: Conjecture (C) If F Is Keller Then Imentioning

confidence: 99%

An ideal-theoretic approach to Keller maps

Miranda-Neto¹

2019

Proceedings of the Edinburgh Mathematical Society

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“…Let l ∈ N and let (P, Q) ∈ L (l) be an (m, n)-pair (see [5,Definition 4.3]). In this section we take (ρ, σ) ∈](0, −1), (1,1)] such that 1 m en ρ,σ (P ) = 1 n en ρ,σ (Q) =: (a/l, b) and a/l > b > 0…”

Section: General Lower Side Cornersmentioning

confidence: 99%

“…When we translated the result and the proofs of [14] into the language of [5], we obtained the same formula for I M , but for I m we obtained only an inequality, consequently we cannot discard the infinite families as desired.…”

Section: Introductionmentioning

confidence: 99%

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The Jacobian Conjecture: Approximate roots and intersection numbers

Guccione,

Horruitiner

et al. 2017

Preprint

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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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On the shape of possible counterexamples to the Jacobian Conjecture

Cited by 14 publications

References 12 publications

An ideal-theoretic approach to Keller maps

An ideal-theoretic approach to Keller maps

The Jacobian Conjecture: Approximate roots and intersection numbers

Jacobian conjecture in $\mathbb R^2$

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