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The Necessary and Sufficient Conditions for the Real Jacobian The necessary and sufficient conditions for the real Jacobian conjecture Yuzhou Tian and Yulin Zhao∗ School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, P.R. China Abstract We focus on investigating the real Jacobian conjecture. This conjecture claims that if F = f 1,...,f n : Rn → Rn is a polynomial map such that det DF is nowhere zero, then F is a global injective. 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 necessary and sufficient condition for the validity of the two-dimensional real Jacobian conjecture, which is an algebraic criterion. By definition a criterion function, F is a global injective if and only if the limit of criterion function is infinite as |x| + |y| tends to infinity. This algebraic criterion improves the main result of Braun et al [J. Differential Equations 260 (2016) 5250-5258]. In the second part, the necessary and sufficient conditions on the n-dimensional real Jacobian conjecture is obtained. Using the tool from the nonlinear functional analysis, F is a global injective if and only if k F (x) k approaches to infinite as k x k→ ∞, which is a generalization of the above algebraic criterion. As an application, we give an alternate proof of the Cima’s result on the n-dimensional real Jacobian conjecture [Nonlinear Anal. 26 (1996) 877-885]. 2020 Math Subject Classification: Primary 14R15. Secondary 08B30. Tertiary 34C05 Key words and phrases: Real Jacobian conjecture; Monodromy; Bendixson compactification; Criterion function arXiv:2011.11843v2 [math.DS] 16 Jan 2021 1 Introduction and main results Let F (x) = f 1 (x) ,...,f n (x) : Rn → Rn be a smooth map with the Jacobian determinant n det DF (x) 6= 0 for all x = (x1,...,xn) ∈ R . Obviously, the map F is a local diffeomorphism. However, it is not always global injective in Rn. Actually, one can impose suitable conditions to guarantee that F is a global diffeomorphism, see for example [7, 11, 18, 31] and references therein. In algebraic geometry, the well-known Jacobian conjecture is to state that if F : Cn → Cn is a polynomial map with det DF a non-zero constant, then F is a global injective. This conjecture was first introduced by Keller in 1939, and up to now it is still open problem. Smale [37] 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, 26, 29, 33, 36, 42], etc. The investigation of Jacobian conjecture leads to a stream of valuable results concerning polynomial automorphisms, as shown in survey [3] and book [38], etc. ∗Corresponding author. E-mail address: [email protected] (Y. Tian), [email protected] (Y. Zhao). 1 From now on, we consider the polynomial map F defined in Rn. Another famous conjecture, the real Jacobian conjecture claims that if F : Rn → Rn is a polynomial map with nonvanishing Jacobian determinant, then F is a global injective, see [32]. Unfortunately, this conjecture is false. In 1994, Pinchuk [30] provided a counterexample which is a non-injective polynomial map F in R2 with nonvanishing Jacobian determinant. Nevertheless, the real Jacobian conjecture has still attracted the interest of numerous mathematicians, especially exploring conditions such that this conjecture holds. Based on the structure of polynomial map F , the authors in [9, 10] give sufficient conditions. Gwoździewicz in [22] obtained that the two-dimensional real Jacobian conjecture (i.e., the polynomial map F = (f, g)) 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. In the first part of this paper, we study the two-dimensional real Jacobian conjecture. To de- scribe our main results, we first introduce some notation. Consider a two-dimensional autonomous differential system x˙ = P (x,y) , y˙ = Q (x,y) . (1.1) Denoted by X = (P,Q) the vector field associated to system (1.1). The vector field X is Ck with k ∈ N+ ∪ {∞} if P (x,y) and Q(x,y) are Ck. Let U be an open set of R2. A non-locally constant function H : U → R is called a first integral of X if it is constant along any solution curve of X contained in U. We denote by b (X ) the Bendixson compactification (see subsection 2.3) of vector field X . In particular, if P (x,y) ,Q (x,y) are real polynomials in variables x and y with d = max{degP, degQ}, the expression of b (X ) is given by d u v u v u˙ = u2 + v2 v2 − u2 P , − 2uvQ , , u2 + v2 u2 + v2 u2 + v2 u2 + v2 d u v u v v˙ = u2 + v2 u2 − v2 Q , − 2uvP , , u2 + v2 u2 + v2 u2 + v2 u2 + v2 see subsection 2.3 for more details. Let q ∈ R2 be a singular point of an analytic vector field X in R2. The singular point q is monodromic if there exists a neighborhood of q such that the orbits of the vector field turn around q either in forward or in backward time. We say that the singular point q is a center if there is a neighborhood of q which is filled up with periodic orbits. The period annulus of the center q is the maximal neighbourhood U of q such that all the orbits contained in U \ {q} are periodic. The center q is a global center if its period annulus is the whole R2. A singular point q is called a focus if all orbits in a neighborhood of q spirally approach this singular point either in forward or in backward time. Sabatini [35] gave the following dynamical result. Theorem 1. Let F = (f (x,y) , g (x,y)) : R2 → R2 be a polynomial map with nowhere zero Jacobian determinant such that F (0, 0) = (0, 0). Then the following statements are equivalent. (a) The origin is a global center for the Hamiltonian polynomial vector field X , (−ffy − ggy,ffx + ggx) . (1.2) (b) F is a global diffeomorphism of the plane onto itself. This theorem provides a global dynamical condition such that F is a global injective. Recently, Braun and Llibre proved in [6] that if the homogeneous terms of higher degree of f and g do not have real linear factors in common and degf = degg, then F is a global injective. Later on, Braun et al. [5] improved this result in the following theorem. 2 Theorem 2. Let F = (f, g) : R2 → R2 be a polynomial map such that det DF (x,y) is nowhere zero and F (0, 0) = (0, 0). If the higher homogeneous terms of the polynomials ffx + ggx and ffy + ggy do not have real linear factors in common, then F is a global injective. Itikawa et al. [25] give two new classes of polynomial maps satisfying the real Jacobian con- jecture in R2. A new proof of Pinchuk map which is a non-injective can be found in [2]. In these works, their proofs rely only on the qualitative theory of planar differential systems, following ideas inspired by Theorem 1. We note that the essential of their proofs are to characterize the global dynamical behavior of Hamiltonian polynomial vector field X . As we know, the global dynamical analysis of Hamiltonian polynomial vector field in many cases are hard and tedious, because we need to get the local behavior at the all finite and infinite singular points, and to determine their separatrix configurations. For this reason, it is natural to ask whether there exist local dynamical conditions to ensure that F is a global injective in R2. Our first result of this paper provides necessary and sufficient conditions for the validity of the two-dimensional real Jacobian conjecture. Theorem 3. Let F = (f, g) : R2 → R2 be a polynomial map with nowhere zero Jacobian determi- nant such that F (0, 0) = (0, 0). Denote by b (X ) the Bendixson compactification of Hamiltonian vector field X defined in (1.2). Then the following statements are equivalent. (a) F is a global diffeomorphism of the plane onto itself. (b) The origin of the polynomial vector field b (X ) is a center. (c) The origin of the polynomial vector field b (X ) is a monodromic singular point. (d) The origin of the polynomial vector field b (X ) has no hyperbolic sectors. (e) The polynomial vector field b (X ) has a Ck first integral with an isolated minimun at the origin, where k ∈ N+ ∪ {∞}. Remark 1. For the condition (a) of Theorem 1, the origin of X must be a global center.
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