The Jacobian conjecture: Reduction of degree and formal expansion of the inverse

Bass, Hyman; Connell, E. H.; Wright, David L.

doi:10.1090/s0273-0979-1982-15032-7

Cited by 610 publications

(516 citation statements)

References 18 publications

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“…is equal to 1 is injective; see, e.g., [2]. This is an open problem, but for any given dimension n and for any given degree d of the polynomial, the validity of the corresponding case of this conjecture can be resolved by applying the Tarski-Seidenberg algorithm.…”

Section: Remarkmentioning

confidence: 99%

Algorithmics of Checking whether a Mapping Is Injective, Surjective, and/or Bijective

Balreira

Kosheleva

Kreinovich

2014

Constraint Programming and Decision Making

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Abstract. In many situations, we would like to check whether an algorithmically given mapping f : A → B is injective, surjective, and/or bijective. These properties have a practical meaning: injectivity means that the events of the action f can be, in principle, reversed, while surjectivity means that every state b ∈ B can appear as a result of the corresponding action. In this paper, we discuss when algorithms are possible for checking these properties.

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Section: Remarkmentioning

confidence: 99%

Algorithmics of Checking whether a Mapping Is Injective, Surjective, and/or Bijective

Balreira

Kosheleva

Kreinovich

2014

Constraint Programming and Decision Making

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“…To see why the ideal c is tempting to work with, we need to recall a theorem, first conjectured by Wang in the quadratic case [38], and proved in full generality by O. Gabber (see [10]). …”

Section: Iii2 Chains And/or Loops?mentioning

confidence: 99%

“…In fact, this seemingly simple problem is quite an embarrassment. Indeed, some faulty proofs have even been published (see the indispensable [10] and [17] for a review). We will show here that the Jacobian conjecture can be formulated in very nice way as a question in preturbative quantum field theory (QFT).…”

Section: Introductionmentioning

confidence: 99%

The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

Abdesselam¹

2003

Ann. Henri Poincaré

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“…By the chain rule for Jacobians, invertible polynomial maps are Keller maps. The famous Jacobian conjecture states that if char K = 0, then any Keller map is invertible (see, e.g., [1] or [4]). …”

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confidence: 99%

Untitled

2001

Proc. Amer. Math. Soc.

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Abstract. In this paper it is proved that a power linear Keller map of dimension three over a field of characteristic zero is linearly triangularizable.Let K be a field. A polynomial map F in dimension n over K is an n-tupleIf G is another polynomial map of the same dimension, then the composition of F and G is defined byThe polynomial map F is invertible if there exists a polynomial map G such that F •G and G•F are both identities. It is Keller if the determinant of its Jacobian is a nonzero element in K, i.e., det JF ∈ K * . By the chain rule for Jacobians, invertible polynomial maps are Keller maps. The famous Jacobian conjecture states that if char K = 0, then any Keller map is invertible (see, e.g., [1] or [4]).A polynomial map F is power linear if it is of the form (It is cubic linear if it is power linear where d i = 3 for all i. Druzkowski [2] showed that in the case char K = 0 if cubic linear Keller maps are invertible, then the Jacobian conjecture would be true. A polynomial map is triangular if it is of the form It is linearly triangularizable if there exists a linear invertible polynomial mapIt is tame if it can be written as a composition of invertible linear maps and elementary maps. It is not hard to see that linearly triangularizable maps are tame and tame maps are invertible. The tame generators conjecture asserts that an invertible polynomial map is tame. This is proved in dimension two by Jung [5] and van der Kulk [6]. For dimensions beyond two, it is an open problem.In this note we prove that power linear Keller maps in dimension three over a field of characteristic zero are linearly triangularizable (hence tame and invertible) proving both the Jacobian and the tame generators conjecture in this case. (It is worth noting that if the degrees of the components of these maps are all equal to three, then the result is a special case of that of Wright [7] which states that all cubic homogeneous polynomial maps are linearly triangularizable.)

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The Jacobian conjecture: Reduction of degree and formal expansion of the inverse

Cited by 610 publications

References 18 publications

Algorithmics of Checking whether a Mapping Is Injective, Surjective, and/or Bijective

Algorithmics of Checking whether a Mapping Is Injective, Surjective, and/or Bijective

The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

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