The Jacobian Conjecture, a Reduction of the Degree to the Quadratic Case

We define in this paper a class of three indices tensor models, endowed with O(N ) ⊗3 invariance (N being the size of the tensor). This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model (and hence of the colored model) and the U(N ) invariant models. We first exhibit the existence of a large N expansion for such a model with general interactions. We then focus on the quartic model and we identify the leading and next-to-leading order (NLO) graphs of the large N expansion. Finally, we prove the existence of a critical regime and we compute the critical exponents, both at leading order and at NLO. This is achieved through the use of various analytic combinatorics techniques.

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“…Remark. Tree structures can naturally be enumerated by means of quantum field theory techniques [6,42,43].…”

Section: A Explicit Counting Of Melonic Graphsmentioning

confidence: 99%

O(N) Random Tensor Models

2016

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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

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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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“…This represents a reduction theorem to the quadratic case for Jacobian Conjecture, up to the addition of a new parameter, related to the introduction of additional intermediate fields σ. An algebraic proof (not using any QFT arguments) of the Theorem 1.6 was also derived in the original article [dGST16].…”

Section: Rivasseau Modelmentioning

confidence: 99%

“…Let us now state the reduction theorem we will review in this paper: dGST16]). For n ∈ N and d ≥ 3, there exists an injective map Φ :…”

Section: Introductionmentioning

confidence: 99%

Combinatorial quantum field theory and the Jacobian conjecture

Tanasa

2020

Preprint

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In this short review we first recall combinatorial or (0−dimensional) quantum field theory (QFT). We then give the main idea of a standard QFT method, called the intermediate field method, and we review how to apply this method to a combinatorial QFT reformulation of the celebrated Jacobian Conjecture on the invertibility of polynomial systems. This approach establishes a related theorem concerning partial elimination of variables that implies a reduction of the generic case to the quadratic one. Note that this does not imply solving the Jacobian Conjecture, because one needs to introduce a supplementary parameter for the dimension of a certain linear subspace where the system holds.

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The Jacobian Conjecture, a Reduction of the Degree to the Quadratic Case

Cited by 7 publications

References 16 publications

O(N) Random Tensor Models

O(N) Random Tensor Models

The necessary and sufficient conditions for the real Jacobian conjecture

Combinatorial quantum field theory and the Jacobian conjecture

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