The Necessary and Sufficient Conditions for the Real Jacobian

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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 describe 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 conjecture 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 determinant 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.

(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. For the condition (b) of Theorem 3, the origin of b (X ) is a local center, not necessary global center. The above conditions (c) and (d) are also local. The last condition (e) is from the point of view of integrability. The vector field b (X ) always exists a first integral (see Section 4), but such a first integral in general cannot be extended to the origin of b (X ). Theorem 3 implies that the local dynamical behavior of b (X ) at the origin determines fully whether polynomial map F is a global injective. So it allows us to investigate real Jacobian conjecture by some elementary dynamical tools which is the local analysis of singular points, for example blow up technique. In fact, the origin of vector field b (X ) is degenerate (see proof of Theorem 3), that is, its linear part identically zero. The two classical problems for degenerate singular point are respectively monodromy problem to decide whether it is of focus-center type, and stability problem to distinguish between a center and a focus. For degenerate singular point, these two problems are very complicated in general vector field, see [19-21]. It is worth to notice that the monodromy and stability problems of vector field b (X ) at the origin are completely solved under the assumption of Theorem 3. Using Theorem 3, we present a necessary and sufficient condition such that the two-dimensional real Jacobian conjecture holds, which is an algebraic criterion. Our second result is described as follows.

Theorem 4. Let F = (f, g) : R2 → R2 be a polynomial map with nowhere zero Jacobian determinant such that F (0, 0) = (0, 0). Deﬁne a polynomial criterion function as follows:

I (x,y)= f 2 (x,y)+ g2 (x,y) . (1.3)

Then F is a global injective if and only if

lim I(x,y)=+∞. (1.4) |x|+|y|→+∞

3 The following example [10] shows that Theorem 4 improves Theorem 2.

Example 1. Consider the polynomial map F = (f, g) : R2 → R2 with f = y +y3 and g = x+xy2. Here det DF = − 1+ y2 1 + 3y2 . The higher homogeneous terms of

2 4 2 2 3 2 2 ffx + ggx = x + 2xy + xy , ffy + ggy = y + 2y x + 2y + y 2x + 3y have y as a common factor. This map F does not satisfy the condition of Theorem 2. The criterion function is given by 2 I(x,y)= x2 + y2 1+ y2 ≥ x2 + y2.

Obviously, lim|x|+|y|→+∞ I(x,y)=+∞. So F is a global injective. After completing the proof of Theorem 4, a natural idea appears for us, i.e. how to generalize it from R2 to Rn. Along the method to prove Theorem 4, we attempt to generalize it. There are essential diﬀerences between the dynamic properties of two-dimensional and high-dimensional. Our accidental meeting the tool from nonlinear functional analysis led to the following result.

Theorem 5. Let F = (f 1,...,f n) : Rn → Rn be a polynomial map with nowhere zero Jacobian determinant such that F (0,..., 0) = (0,..., 0). Then F is a global injective if and only if

lim k F (x) k= ∞. (1.5) kxk→∞

Here, k · k deﬁnes a norm on Rn. Remark 2. Although Theorem 4 is a consequence of Theorem 5, we must once again emphasize that our approach to prove Theorem 4 is based on qualitative theory of dynamical systems, and is completely diﬀerent from the proof of Theorem 5. We provide a very elementary dynamical proof.

The polynomial R (x1,...,xn) is quasi-homogeneous of weighted degree l with respect to weight exponents s = (s1,...,sn) if there exist positive integers s1,...,sn and l such that for arbitrary λ ∈ + s1 sn l R = {λ ∈ R,λ> 0}, R (λ x1, . . . , λ xn)= λ R (x1,...,xn). To each polynomial R (x1,...,xn), d it can be written as the sum of its quasi-homogeneous parts R = i=0 Ri, where Ri is quasi- homogeneous polynomial of weighted degree i with respect to weight exponents s. Moreover, the P quasi-homogeneous term Rd is called the higher quasi-homogeneous term of polynomial R with n n respect to weight exponents s, and denote by Rs. For a polynomial map F = (f 1,...,f ) : R → n 1 n R , we denote Fs = fs ,...,fs . With the help of the algebraic skills, Cima et al. in [10] proved the following theorem. Theorem 6. Let F = (f 1,...,f n) : Rn → Rn be a polynomial map with nowhere zero Jacobian determinant such that F (0,..., 0) = (0,..., 0). If there is a weight exponents s = (s1,...,sn) ∈ n N+ such that Fs(x)= 0 has only the trivial solution x = 0, then F is a global injective. By Theorem 5, we give an alternate proof of Theorem 6. Our method to prove Theorem 6 is completely diﬀerent from [10]. Note that Theorem 6, Cima’s result, is not a necessary and suﬃcient condition. In the above 3 2 Example 1, Fs for any weight exponents s = (s1,s2) is always y ,xy , which has nontrivial real solutions. This shows that Theorem 5 improves Cima’s result, i.e., Theorem 6. The rest of this paper is organized as follows. In section 2 we present some preliminary results. Section 3 is devoted to prove Theorem 3. Theorem 4 is proved in section 4. The proofs of Theorems 5 and 6 will be given in section 5.

2 Preliminary results

In this section, we introduce some preliminary results.

4 2.1 Limit sets

Let ϕ (t,p) be the integral curve of system (1.1) passing through the point p ∈ R2 such that ϕ (0,p) = p. The set K is positively invariant if for each p ∈ K, ϕ (t,p) ∈ K for all t ≥ 0. We deﬁne the following sets

2 ω (p)= x0 ∈ R : there exist {tn} with tn →∞ and ϕ (tn,p) → x0 when n →∞ and

2 α (p)= x0 ∈ R : there exist {tn} with tn → −∞ and ϕ (tn,p) → x0 when n →∞ . The sets ω (p)and α (p) are called the ω-limit set and the α-limit set of p, respectively. Note that an α-limit set of an integral curve ϕ (t,p) is the ω-limit set of the integral curve ϕ (−t,p), i.e., after the time reversal. For this reason, it is suﬃcient to study the ω-limit sets. The following theorem characterized the structure of ω-limit sets, see [16] or [39]. Theorem 7 (Poincaré-Bendixson Theorem). Let K be a positively invariant compact set for system (1.1) containing a ﬁnite number of singular points, and p ∈ K. Then one of the following statements holds. (a) ω (p) is a singular point.

(b) ω (p) is a periodic orbit.

(c) ω (p) consists of a finite number of singular points p1,...,pn and a finite number of orbits γ1, . . . , γn such that α (γi)= pi, ω (γi)= pi+1 for i = 1,...,n−1, α (γn)= pn and ω (γn)= p1. Possibly, some of the singular points pi are identified.

2.2 Local structure of isolated singular points To characterize the local structure of isolated singular point q of analytic vector field X , we need the following definitions, see [16, 40, 41] for more details. Let U be a local region limited by two orbits of X inside with vertex singular point q. The local region U is a hyperbolic sector of singular point q if all orbits resemble hyperbolae in U, see (1) of Figure 1. The local region U is an elliptic sector of singular point q if all orbits have the singular point q as both α and ω limit sets, see (2) of Figure 1. The local region U is a parabolic sector of singular point q if all orbits positively (or all negatively) flow into singular point q, see (3) of Figure 1. Note that the two boundaries of a hyperbolic sector are separatrices of singular point q.

PSfrag replacements PSfrag replacements PSfrag replacements q q q (1) Hyperbolic sector (2) Elliptic sector (3) Parabolic sector Figure 1: Sectors near a singular point q.

The next theorem is proved in [23, 27]. Theorem 8. If q is an isolated singular point of analytic vector ﬁeld X , then singular point q can only be a center, a focus, or be decomposed into a ﬁnite number of hyperbolic, elliptic and parabolic sectors.

5 2.3 Compactification of vector field If P (x,y) ,Q (x,y) are real polynomials in variables x and y, then system (1.1) is polynomial system. We say that system (1.1) has degree d if d = max{degP, degQ}. In order to investigate the behavior of the trajectories of a planar polynomial vector field X near infinity, we need to compactify it. Usually there are two methods to compactify a planar polynomial vector field: Pioncaré compactification and Bendixson compactification. Next we introduce these two compactifications, see Chapter 13 of [1] or Chapter 5 of [16] for more details. We first briefly describe the Pioncaré compactification. The (Poincaré) unit sphere is tangent to xy-plane at the origin (0, 0) in Figure 2. The point M(x,y) in the xy-plane connects with the center O of the sphere through a straight line which intersects the sphere at the two points M ′ and M ′. We project the point M ′ on the lower hemisphere vertically in to the xy-plane which leads to the point M ′′ on the Poincaré disk D2 = (x,y) x2 + y2 ≤ 1 . It is known that the boundary of the disc D2, i.e. the unit circle S1 = (x,y) x2 + y2 = 1 , corresponds to the infinity of R2, and called the equator. Then the vector field X can be extended analytically to Poincaré sphere by the central projection. So the global dynamics of X can be characterized on the Poincaré disk, that is, the finite and infinity of X respectively corresponding the interior and boundary S1 of D2.

M′

1 Equator S

PSfrag replacements ′ y M

2 Poincaré disc D M(x, y)

Figure 2: The Pioncaré compactiﬁcation and Poincaré disc.

Roughly speaking, the construction of the Bendixson compactification is as follows. Let S2 = 3 2 2 2 X {Y = (y1,y2,y3) ∈ R : y1 + y2 + y3 = 1/4} (the Bendixson sphere). Assume that is defined with the tangent plane to the sphere S2 at the south pole S = (0, 0, −1/2), that is, xy-plane, see Figure 3. The Bendixson compactified vector field b (X ) associated to X is an analytic vector field induced on S2 by the stereographic projection. More precisely, consider the stereographic projection pN from the north pole N = (0, 0, 1/2) to the xy-plane. Thus the vector field X can 2 −1 be induced to S \ N by the map pN . Obviously, the infinity of the xy-plane is transformed by −1 pN into the north pole N. To simplify the calculations, we take the two local charts on the Bendixson sphere S2 given by

2 2 UN = S \ N,US = S \ S with associated local maps 2 2 pN : UN → R , pS : US → R , 2 where pS is the stereographic projection of S from the south pole S to the uv-plane given by the −1 equation y3 = 1/2. The map pS ◦ pN from the xy-plane minus S to the uv-plane minus N is given by x y u = , v = . (2.1) x2 + y2 x2 + y2

6 y3

uv-plane N

O PSfrag replacements y2

S xy-plane

Figure 3: The stereographic projection and Bendixson sphere.

After a scaling of the independent variable in the local chart (UN ,pN ) the expression for b (X ) is

d u v u v u˙ = u2 + v2 v2 − u2 P , − 2uvQ , , u2 + v2 u2 + v2 u2 + v2 u2 + v2  (2.2)  d u v u v v˙ = u2 + v2 u2 − v2 Q , − 2uvP , . u2 + v2 u2 + v2 u2 + v2 u2 + v2  Hence the inﬁnity of system (1.1) becomes the origin of system (2.2).

Remark 3. By the Bendixson compactification, the infinity of system (1.1) on the Poincaré disk, i.e the equator S1, is transformed into the origin of system (2.2). We say in what follows, unless otherwise specified, that the infinity of a vector field is refer to the equator S1 on the Poincaré disk.

2.4 Topological index The following results are well known, see Chapter 6 of [16] or [41].

Theorem 9 (Poincaré Index Formula). Let q be an isolated singular point having the ﬁnite secto- rial decomposition property. Let e, h and p denote the number of elliptic, hyperbolic and parabolic sectors of q, respectively. Then the index of q is (e − h) /2 + 1.

Proposition 1. If a vector ﬁeld X has only isolated singular point, then the sum of indices singular points in a region D enclosed by its any periodic orbit is 1.

The next proposition follows immediately from Theorem 9.

Proposition 2. For an analytic vector ﬁeld X , the index of a monodromic singular point (i.e., a center or a focus) is 1.

3 Proof of Theorem 3

The main purpose of this section is to prove Theorem 3.

Lemma 1. Let F = (f, g) : R2 → R2 be a polynomial map with nowhere zero Jacobian determinant such that F (0, 0) = (0, 0). If gcd(f, g) is the greatest common divisor of f and g, then y ∤ gcd(f, g) and x ∤ gcd(f, g).

7 Proof. Suppose that y | gcd(f, g). Then there exist two polynomials f¯1 (x,y) and g¯1 (x,y) such ¯ that f = yf1 and g = yg¯1. This is in contraction with detDF (0, 0) 6= 0. Thus, y ∤ gcd(f, g). By a similar way, one can prove that x ∤ gcd(f, g) also holds. The following theorem is due to Mazzi and Sabatini [28].

Theorem 10. Assume that system (1.1) is Ck with an isolated singular point q and k ∈ N+ ∪{∞}. Then the singular point q is a center if and only if there exists a Ck ﬁrst integral with an isolated minimum at q.

Proposition 3. Let F = (f, g) : R2 → R2 be a polynomial map with nowhere zero Jacobian determinant such that F (0, 0) = (0, 0). Then q is a singular point of the Hamiltonian vector ﬁeld (1.2) if and only if F (q) = (0, 0). In this case, this singular point q is a center of vector ﬁeld (1.2).

Proof. The Hamiltonian vector ﬁeld X can be written as

x˙ −f −g f = y y . y˙ f g g x x Indeed, q is a singular point of X if and only if

−f (q) −g (q) f (q) 0 y y = . f (q) g (q) g (q) 0 x x The suﬃciency is obvious. Since det DF (q) 6= 0, f (q)= g (q) = 0. The necessity holds. Since det DF (q) 6= 0, singular point q is an isolated. The Hamiltonian vector ﬁeld (1.2) has the Hamiltonian f 2 (x,y)+ g2 (x,y) H (x,y)= . (3.1) 2 Clearly, q is an isolated minimum of H, that is, H (x,y) ≥ H (q) = 0. By Theorem 10, q is a center. This proves the Proposition 3.

n m Lemma 2. Let f (x,y) = i=1 fi (x,y), g (x,y) = j=1 gj (x,y) and F = (f, g) with nowhere zero Jacobian determinant det DF (x,y), where fi (x,y) and gj (x,y) are homogeneous polynomials P P of degree i and j, respectively. Then f1 (x,y)= g1 (x,y) = 0 if and only if (x,y) = (0, 0).

Proof. Since f1 (x,y) and g1 (x,y) = 0 are linear, we have

f (x,y) f f x 0 1 = 1x 1y = . (3.2) g (x,y) g g y 0 1 1x 1y The determinant of the coeﬃcient matrix of system (3.2) is det DF (0, 0) 6= 0. The lemma holds.

Lemma 3. Let φ (t,p) be the trajectory of vector ﬁeld (1.2) passing through the regular point p ∈ R2 for t ∈ R. Then one of the following statements holds. (a) ω (p)= α (p) is a periodic orbit located at a period annulus.

(b) ω (p) ⊂ S1 and α (p) ⊂ S1.

Here S1 is the infinity of the Poincaré disc. Proof. By Proposition 3, each finite singular point q of vector field (1.2) is a center. From Theorem 7 it follows that ω (p)= α (p) is a periodic orbit, or ω (p) ⊂ S1 and α (p) ⊂ S1. The limit sets of vector field b (X ) are given in next proposition.

8 Proposition 4. Let ψ (t, p¯) be the trajectory of vector ﬁeld b (X ) passing through the regular point p¯ ∈ R2 for t ∈ R. Then one of the following statements holds. (a) ω (¯p)= α (¯p) is a periodic orbit located at a period annulus.

(b) ω (¯p)= α (¯p) is the origin of vector ﬁeld b (X ) .

Proof. By Lemma 3, on the Bendixson sphere S2 (see Figure 3) the α and ω limit sets of each orbit of the vector ﬁeld b (X ) can only be the north pole N or a periodic orbit located at a period annulus. This proposition is conﬁrmed.

n m Let f (x,y)= i=1 fi (x,y), g (x,y)= j=1 gj (x,y) and d = max{n,m}, where fi (x,y) and gj (x,y) are homogeneous polynomials of degree i and j, respectively. From the equation (2.2), P P the Bendixson compactiﬁcation of vector ﬁeld X is given by

d d 2 2 2d−i−j 2 2 u˙ = u + v [ u − v (fi (u, v) fjy (u, v)+ gi (u, v) gjy (u, v))  i=1 j=1  P P  −2uv (fi (u, v) fjx (u, v)+ gi (u, v) gjx (u, v))],  (3.3)  d d  2 2 2d−i−j 2 2 v˙ = u + v [ u − v (fi (u, v) fjx (u, v)+ gi (u, v) gjx (u, v)) i=1 j=1  P P  +2uv (fi (u, v) fjy (u, v)+ gi (u, v) gjy (u, v))].   It is easy to check that the origin of system (3.3) is degenerate. The local dynamical behavior of vector ﬁeld b (X ) on the Poincaré disk is given as follows.

Proposition 5. 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). Then the following statements hold.

(a) The ﬁnite singular points of polynomial vector ﬁeld b (X ) other than its origin are centers.

(b) For polynomial vector ﬁeld b (X ), there are no inﬁnite singular points on the Poincaré disc.

Proof. (a) By Proposition 3, statement (a) can be easily proved. (b) Taking the Poincaré transformation u = 1/v¯, v =u/ ¯ v¯ and rescaling the time dt =v ¯4d−1dτ, system (3.3) can be written as

d d 2 i+j−2 2 2d−i−j u¯˙ = 1 +u ¯ jv¯ 1 +u ¯ (fi (1, u¯) fj (1, u¯)+ gi (1, u¯) gj (1, u¯)) ,  i=1 j=1 P P  d d (3.4)  ˙ i+j−2 2 2d−i−j 2 v¯ = −v¯ v¯ 1 +u ¯ [ 1 − u¯ (fi (1, u¯) fjy (1, u¯)+ gi (1, u¯) gjy (1, u¯))  i=1 j=1 P P  −2¯u (fi (1, u¯) fjx (1, u¯)+ gi (1, u¯) gjx (1, u¯))].   Applying Lemma 2, we have that the system (3.4) has no singular points on the u¯-axis. Using the Poincaré transformation u =u/ ¯ v¯, v = 1/v¯ with the scaling dt =v ¯4d−1dτ, system (3.3) becomes

d d 2 i+j−2 2 2d−i−j u¯˙ = − 1 +u ¯ jv¯ 1 +u ¯ (fi (¯u, 1) fj (¯u, 1) + gi (¯u, 1) gj (¯u, 1)) ,  i=1 j=1 P P  d d (3.5)  ˙ i+j−2 2 2d−i−j 2 v¯ = −v¯ v¯ 1 +u ¯ [ u¯ − 1 (fi (¯u, 1) fjx (¯u, 1) + gi (¯u, 1) gjx (¯u, 1))  i=1 j=1 P P  +2¯u (fi (¯u, 1) fjy (¯u, 1) + gi (¯u, 1) gjy (¯u, 1))].   Similarly, the origin is not a singular point of system (3.5). The statement (b) holds.

9 (c) Assume that the origin of b (X ) has a attracting parabolic sector V . Let ψ (t, p¯) be the trajectory of vector ﬁeld b (X ) passing through the regular point p¯ ∈ V . Then ω (¯p) is the origin of b (X ). From Proposition 4, α (¯p) is also the origin of b (X ), which means that V is an elliptic sector. This is a contradiction. So statement (c) is conﬁrmed.

Proof of Theorem 3. For clarity, we will split the proof into three steps. Firstly, we prove that (a) ⇒ (b) ⇒ (c) ⇒ (d). Using Theorem 1, it is easy to prove that (a) ⇒ (b). By the definition of monodromic singular point, it is obvious that (b) ⇒ (c) ⇒ (d). Secondly, we show that (d) ⇒ (c) ⇒ (b) ⇒ (a) as follows. (d) ⇒ (c). By Theorem 9, the index of the origin of the polynomial vector field b (X ) is e/2+1. Since the origin of vector field X is a center, on the Bendixson sphere S2 (see Figure 3) the south pole S is also a center. This means that there exists a periodic orbit of b (X ) such that it contains all finite singular points of b (X ). Let c¯ denote the number of finite singular points of b (X ) other than its origin. By statement (a) of Proposition 5 and Proposition 1, we have e/2+1+¯c = 1, that is, e =c ¯ = 0. So, the origin is the unique finite singular point of b (X ). From statement (c) of Proposition 5 and Theorem 8 it follows that the origin is a monodromic singular point. (c) ⇒ (b). I’lyashenko in [24] and Écalle in [17] prove that a monodromic singular point of an analytic vector field must be either a center or a focus. Assume that the origin of b (X ) is a attracting focus. Let U be a sufficiently small neighborhood of the origin. Then there exists a orbit ψ (t, p¯) of b (X ) passing through the regular point p¯ ∈ U \ {O} such that limt→+∞ ψ (t, p¯) = O, that is, ω (¯p)= {O}. By Proposition 4, α (¯p)= {O} which is a contradiction. Thus the origin of b (X ) is a center. (b) ⇒ (a). The origin of the polynomial vector field b (X ) is a center, which means that every solution trajectory of the polynomial vector field X in a neighborhood of infinity (the equator S1) is a closed orbit. Thus all finite singular points of X are contained in a periodic orbit. From Propositions 1, 2 and 3, it follows that the (0, 0) is the unique finite singular point of X , which is a center. Combining with the behavior of the trajectories of X near infinity, we obtain that (0, 0) is a global center of the vector field X . Using Theorem 1, F is a global diffeomorphism of the plane onto itself. Finally, it follows from Theorem 10 that statement (e) is equivalent to statement (b), that is, (b) ⇔ (e). We complete the proof of Theorem 3.

4 Proof of Theorem 4

Let l be a straight line and a point E ∈ l. The point E is a contact point of the straight line l with a vector field (1.1) if the vector X (E) is parallel to l. To prove Theorem 4, we need the following lemmas. Lemma 4. Assume that vector field (1.1) is a polynomial vector field with d¯ = max {degP, degQ}. Let l be a straight line. Then either l is an invariant straight line of X , or X has at most d¯ contact points (including the singular points) along l. The proof of Lemma 4 can be found in Lemma 8.12 of [16]. 2 Lemma 5. Let G1 (x,y) and G2 (x,y) be two functions defined on R . Then the limit 1 lim (4.1) (x,y)→(0,0) x y x y G2 , + G2 , 1 x2 + y2 x2 + y2 2 x2 + y2 x2 + y2 exists if and only if the limit 1 lim 2 2 (4.2) |x|+|y|→+∞ G1 (x,y)+ G2 (x,y) exists. Moreover, these two limits are equal.

10 Proof. Suﬃciency. Assume that 1 lim 2 2 = L. |x|+|y|→+∞ G1 (x,y)+ G2 (x,y) For every ε> 0, there exists M > 0 such that if |x| + |y| > M, then 1 −L < ε. (4.3) G2 (x,y)+ G2 (x,y) 1 2

Let X Y (x,y)= , . X2 + Y 2 X2 + Y 2 Then 2 X2 + Y 2 2 |X| + |Y |≤ = . |X| + |Y | |x| + |y| Taking δ = 2/M, we have that if 0 < |X| + |Y | < δ, then

1 −L < ε (4.4) 2 X Y 2 X Y G 2 2 , 2 2 + G 2 2 , 2 2 1 X +Y X +Y 2 X +Y X +Y

for every ε> 0. Consequently, the limit (4.1) exists. Necessity. Assume that 1 lim = L1. (x,y)→(0,0) x y x y G2 , + G2 , 1 x2 + y2 x2 + y2 2 x2 + y2 x2 + y2 For every ε> 0, there exists δ1 < 0 such that if 0 < |x| + |y| < δ1, then

1 −L < ε. (4.5) x y x y 1 G2 , + G2 , 1 2 2 2 2 2 2 2 2 2 x + y x + y x + y x + y

Setting x y (X ,Y )= , , 1 1 x2 + y2 x2 + y2 one can obtain that 2 2 X1 + Y1 1 |X1| + |Y1|≥ = . |X1| + |Y1| |x| + |y|

Taking M1 = 1/δ1, we get that if |X1| + |Y1| > M1, then 1 −L < ε (4.6) G2 (X ,Y )+ G2 (X ,Y ) 1 1 1 1 2 1 1

for every ε> 0. So, the limit ( 4.2) exists. This lemma is conﬁrmed.

The Lemma 5 tells us that Theorem 4 is equivalent to the following proposition. Proposition 6. Let F = (f, g) : R2 → R2 be a polynomial map with nowhere zero Jacobian determinant such that F (0, 0) = (0, 0). Then F is a global injective if and only if 1 lim I¯(x,y) = lim (4.7) (x,y)→(0,0) (x,y)→(0,0) x y x y f 2 , + g2 , x2 + y2 x2 + y2 x2 + y2 x2 + y2 exists.

11 n m Proof. Suﬃciency. Let f (x,y) = i=1 fi (x,y), g (x,y) = j=1 gj (x,y) and d = max{n,m}, where fi (x,y) and gj (x,y) are homogeneous polynomials with degree i and j, respectively. Then P2 2 P there exists a θ0 ∈ [0, 2π] such that fd (cos θ0, sin θ0)+gd (cos θ0, sin θ0) 6= 0. Otherwise, fd (x,y)= gd (x,y) ≡ 0. We have 1 lim I¯(r cos θ0,r sin θ0) = lim r→0 r→0 2 cos θ0 sin θ0 2 cos θ0 sin θ0 f r , r + g r , r r2d = lim r→0 d d 2d−i−j P P r [fi (cos θ0, sin θ0) fj (cos θ0, sin θ0)+ gi (cos θ0, sin θ0) gj (cos θ0, sin θ0)] i=1 j=1 = 0. (4.8) Therefore, I¯(x,y) → 0 as (x,y) → (0, 0) along the straight line

l := {(x,y) : x = r cos θ0,y = r sin θ0, r ∈ R} . ¯ Since the limit lim(x,y)→(0,0) I(x,y) exists, we obtain

lim I¯(x,y) = lim I¯(x,y) = 0. (4.9) (x,y)→(0,0) (x,y) → (0, 0) Along straight line l

The Hamiltonian vector ﬁeld (1.2) has the Hamiltonian

f 2 (x,y)+ g2 (x,y) H (x,y)= . (4.10) 2 By Proposition 1.2 of [40], the vector ﬁeld b (X ) has the ﬁrst integral 1 I¯(u, v)= u v u v f 2 , + g2 , u2 + v2 u2 + v2 u2 + v2 u2 + v2 with u2 + v2 6= 0. It is easy to check that

I¯(u, v) , (u, v) 6= (0, 0) , I˜(u, v)= (4.11) (0, (u, v) = (0, 0) , is also ﬁrst integral of vector ﬁeld b (X ). Since

lim I¯(u, v) = 0, (u,v)→(0,0)

I˜(u, v) is continuous on R2. Let γ(t) = (u(t), v(t)) be an orbit of b (X ) tending to the origin as t → +∞ (or t → −∞). There exists a constant C such that I˜(γ(t)) = I˜(u(t), v(t)) = C. Since I˜ is a continuous function on R2, C = limt→+∞ I˜(γ(t)) = limt→+∞ I˜(u(t), v(t)) = 0. This means that γ(t) = (u(t), v(t)) = (0, 0). There are no orbits tending or leaving the origin. Thus, the origin of b (X ) is a monodromic singular point. By statement (c) of Theorem 3, F is a global injective. Necessity. If F is a global injective, then the origin of b (X ) is a center (also a global center). From the equation (4.8), we know that I¯(x,y) → 0 as (x,y) → (0, 0) along the straight line l, that is, limr→0 I¯(r cos θ0, r sin θ0) = 0. For every ε> 0, there exists δ1 > 0 such that if 0

I¯(r cos θ0, r sin θ0) < ε. (4.12)

Applying Lemma 4, there exists a δ2 > 0 such that the line segment OA = {(r cos θ0, r sin θ0) : r ∈ [0, δ2]} ⊂ l is a transverse section of b (X ). Let δ3 = min {δ1/2, δ2/2} and ψ (t, p¯) be the

12 trajectory of vector field b (X ) passing through the point p¯ = (δ3 cos θ0, δ3 sin θ0) for t ∈ R. Then ψ (t, p¯) is a periodic orbit. Let R be a closed region bounded by the ψ (t, p¯). There exists a neighborhood U˚ (O) , (u, v) : 0 < u2 + v2 < δ such that U˚ (O) $ R. By definition of the first integral, we have I¯(u0, v0) = I¯(ψ (t, p¯0)) for each p¯0 = (u0, v0) ∈ U˚ and t ∈ R. Since ψ (t, p¯0) is a periodic orbit and the line segment OA is a transverse section, then there exists time t¯ such that ψ (t,¯ p¯0) intersects OA at the point pˆ0 = (ˆr cos θ0, rˆsin θ0) with 0 < rˆ < δ3, see Figure 4. Using equation (4.12), we get

I¯(u0, v0)= I¯(ψ (t, p¯0)) = I¯(ˆr cos θ0, rˆsin θ0) < ε, that is, I¯(u0, v0) < ε for arbitrary (u0, v0) ∈ U˚. ˚ Based on the analysis above, for every ε > 0, there exists a neighborhood U such that if ˚ ¯ ¯ (u, v) ∈ U, then I (u, v) < ε. Therefore, the lim(u,v)→(0,0) I(u, v) exists. This completes the proof of Proposition 6.

ψ (t, p¯)

ψ (t, p¯0)

R PSfrag replacements O

U˚

Figure 4: The closed region R and neighborhood U˚.

Proof of Theorem 4. The Theorem 4 follows by Proposition 6.

5 Proofs of Theorems 5 and 6

Our main aim of this section is to prove Theorems 5 and to give an alternate proof of Theorem 6. Central to the proof of the Theorem 5 is the following lemma, see Exercise 4 of page 171 of [13] or Theorem 2.2 of [34]

Lemma 6. Let F : Rn → Rn be a C1-map and detDF (x) 6= 0 on Rn. Then F is a homeomor- phism onto Rn if and only if k F (x) k→ ∞ as k x k→ ∞. Proof of Theorem 5. The Theorem 5 holds immediately by Lemma 6.

The rest of this section is to present an alternate proof of Theorem 6.

n n 2 Lemma 7. Let s = (s1,...,sn) ∈ N+ and j=1 yj = 1. If

yP1 yn x1 = ,...,xn = , rs1 rsn

n 2 then j=1 xj →∞ if and only if r → 0. P

13 Proof. Without loss of generality, we can assume s1 ≥···≥ sn. For 1 >r> 0, one can obtain that n 1 2 1 ≥ xj ≥ . r2s1 r2sn j X=1 From this inequation, the lemma follows.

Proof of Theorem 6. By Theorem 5, it is enough to prove that

lim k F (x) k= ∞. kxk→∞

For convenience, the norm k · k here can take square norm due to the fact that the norms in ﬁnite i mi i i dimensional space are all equivalent. Let f (x) = l=1 fl (x) with i = 1,...,n, where fl (x) is quasi-homogeneous of weighted degree l with respect to weight exponents s = (s1,...,sn). Here, n P s x 1 F ( )= fm1 ,...,fmn . Without loss of generality, we can assume m ≥···≥ m . Consider the change of coordinates 1 n y1 yn x1 = ,...,xn = rs1 rsn n−1 with r> 0 and y = (y1,...,yn) ∈ S = {y :k y k= 1}. By Lemma 7, we have

1 1 r2m1 lim = lim 2 = lim 2 (5.1) kxk→∞ k F (x) k r→0 n mi r→0 n mi 1 i m1−l i rl fl (y) r fl (y) i=1 l=1 i=1 l=1 P P P P with y ∈ Sn−1. Note that

mi 2 mi 2 m1−l i 2m1−2mi mi−l i r fl (y) = r r fl (y) . (5.2) l ! l ! X=1 X=1 Since

mi 2 mi−l i i 2 lim r fl (y) = fm (y) , r→0 i l=1 ! X there exists a δi > 0 such that

mi 2 1 2 f i (y) < rmi−lf i (y) (5.3) 2 mi l l=1 ! X n−1 for all 0

r2m1 2r2m1 2r2mn 2r2mn 0 < 2 < n ≤ n = n mi 2 2 s y 2m −2mi i i k F ( ) k m1−l i r 1 f (y) f (y) r fl (y) mi mi i=1 l=1 i=1 i=1 P P P P n for all 0 0 such that L ≤k Fs (y) k for all y ∈ S −1. Therefore,

1 2r2mn 0 ≤ lim ≤ lim = 0, that is, lim k F (x) k= ∞. kxk→∞ k F (x) k r→0 k Fs (y) k kxk→∞ This ends the proof.

14 Acknowledgments

The authors would like to thank Professor Changjian Liu for his valuable suggestions and comments (for example, the information on Lemma 6 and the proof details of Theorem 6). This research is supported by the National Natural Science Foundation of China (No.11971495 and No.11801582) and China Scholarship Council (No. 201906380022).

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