A new qualitative proof of a result on the real jacobian conjecture

Abstracts

Let F = (f, g) : R² → R²be a polynomial map such that det DF (x) is different from zero for all x ∈ R². We assume that the degrees of f and g are equal. We denote by the homogeneous part of higher degree of f and g, respectively. In this note we provide a proof relied on qualitative theory of differential equations of the following result: If do not have real linear factors in common, then F is injective.

Real Jacobian conjecture; global injectivity; center; Poincaré compactification

Seja F = (f, g) : R2 → R2 uma aplicação polinomial tal que det DF(x) é diferente de zero para todos x ∈ R2. Assumimos que os graus de f e g são iguais. Denotamos por e as partes homogêneas de maior grau de f e g, respectivamente. Nesta nota, damos uma demonstração baseada na teoria qualitativa de equações diferenciais do seguinte resultado: Se F é injetora. não têm fatores lineares em comum, então

Conjectura Jacobiana real; injetividade global; centro; compactificação de Poincaré

1- INTRODUCTION

Let F = (f, g) : R² → R² be a smooth map such that det DF is nowhere zero. It is clear thatF is a local diffeomorphism, but it is not always injective. There are very general well known conditions to ensure that F is a global diffeomorphism, for instance F is a global diffeomorphism if and only if it is proper (i.e. if inverse images of compact subsets are compact), orF is a diffeomorphism if and only if Plastock 1974). Another condition, now specifically of R2 and ensuring just the injectivity of F, is the following sufficient condition: the real eigenvalues of DF (x), for all x ∈ R2, are not contained in an interval of the form (0, ε), for some ε > 0, see (Fernandes et al. 2004) and (Cobo et al. 2002). These conditions are due to Banach-Mazur and Hadamard, respectively, and remain true in more general spaces, for details, see (

Now, if F is a polynomial map, the statement that F is injective is known as the real Jacobian conjecture. This conjecture is false, since Pinchuk constructed, in (Pinchuk 1994), a non injective polynomial map with nonvanishing Jacobian determinant. So it is natural to ask for additional conditions in order for this conjecture to hold. In (Braun and dos Santos Filho 2010), for example, it is shown that it is enough to assume that the degree of f is less or equal to 3. On the other hand, if we assume that det DF (x) = constant ≠ 0, then to know if F is injective is an open problem largely known as the Jacobian conjecture, see (Bass et al. 1982) and (Van den Essen 2000) for details and for surveys on the Jacobian conjecture.

In the next theorem, we provide another suffi cient condition for the validity of the real Jacobian conjecture. Our result is not new. Indeed it is a consequence of the main result of (Cima et al. 1996). But our proof is a very elementary dynamical one and relies on qualitative theory of differential equations.

Before the statement of the theorem, we introduce some notations. Given a polynomial map p : R2 → R, we write the homogeneous part of higher degree of p. If q : R2 → R is another polynomial map, we say that p and q do not have real linear factors in common when there is not a linear polynomial ax + by which divides both p and q.

Theorem 1. Let F = (f, g) : R2 → R2 be a polynomial map such that det DF is nowhere zero. Assume that the degrees of f and g are equal and greater than one. Then the following condition is sufficient for the global injectivity of F : the homogeneous polynomials and do not have real linear factors in common.

The following map shows that the condition in Theorem 1 is not necessary for global injectivity:

For this polynomial map F1, we observe that the degree off1 and g1 is 3, detDF1(x, y) = 1 + 2(3x +y)2/3 > 0, and the homogeneous polynomials x + y. We recall that in (Braun and dos Santos Filho 2010) it is proved that all the polynomial maps F : R² → R² with one of the components with degree less than or equal to 3 whose Jacobian is nowhere zero, are injective, so F1 is injective. have in common the real linear factor 3

Now we shall show that there are polynomial maps satisfying the as sumption of Theorem 1. We consider the following class of polynomial maps

F₂ = (f ₂, g ₂) = (− y + x k − y k, x + x k + y k) with k ≥ 1 odd.

For these maps the degree of f2 and g2 isk, det DF2(x, y) = 1 +k(x ^k−1 + y ^k−1) + 2k ² x ^k−1 y ^k−1 > 0, and the homogeneous polynomials x −y and x + y, respectively. Hence, by Theorem 1 it follows that the maps F2 are injective. have no common real linear factors, because the unique real linear factors of are

In section 2 we summarize some results that we shall use in the proof of Theorem 1, which will be depicted in section 3. In section 4 we show that the Pinchuk counterexample to the real Jacobian conjecture does not satisfy the assumptions of Theorem 1.

2- PRELIMINARY RESULTS

A singular point p of a differential system defined in R²is a center if it has a neighborhood filled of periodic orbits with the unique exception of p. The period annulus of the center p is the maximal neighborhood P of psuch that all the orbits contained in P are periodic except of course,p.

A center is global if its period annulus is the whole plane R².

Let X be a planar polynomial vector field of degree n. ThePoincaré compactified vector field p(X) corresponding to X is an analytic vector field induced on S² as follows, see for more details (González 1969) or Chapter 5 of (Dumortier et al. 2006).

Let S² = {y = (y1, y2, y3) ∈ R3 : Poincaré sphere) and T _y S² be the tangent space to S² at the pointy. Assume that X is defined in the planeT(0,0,1)S² ≡ R². Consider the central projection f : T(0,0,1)S² → S². This map defines two copies of X, one in the open northern hemisphere H+ and other in the open southern hemisphere H−. Denote by X ' the vector field Df ◦ X defined on S² except on its equator S¹ = {y ∈ S² : y3 = 0}. Clearly S¹ is identified to the infinity of R². In order to extend X ' to a vector field on S²(including S¹⁾ it is necessary that X satisfies suitable conditions. In the case that X is a planar polynomial vector field of degree nthen p(X ) is the only analytic extension of X ' to S² . On S² \ S¹ = H⁺ [ H⁻there are two symmetric copies of X, one in H+ and other in H−, and knowing the behavior of p(X ) around S¹, we know the behavior of X at infinity. The projection of H+ on y3 = 0 under (y1, y2, y3) (y1, y2) is called the Poincaré disc,and it is denoted by D2. The Poincaré compactification has the property that S¹ is invariant under the flow of p(X ). = 1} (the

The singular points of X are called the finite singular points of X or of p(X ). While the singular points ofp(X ) contained in S¹, i.e. at infinity, are called the infinite singular points of X or of p(X ). It is known that the infinity singular points appear in pairs diametrally opposite.

Assume that the two components of the planar polynomial vector field X are the polynomials P and Q, such that nis the maximum of the degrees of P and Q. Denote as usual Pn and Qn the homogeneous parts of degree n of P and Q, respectively. Then it is known that p(X ) has infinite singular points if and only if there exist real linear factors ax +by dividing yPn − xQn, see Chapter 5 of (Dumortier et al. 2006). In this situation, the endpoints of the straight line ax + by= 0 provide the infinite singular points.

Now we assume that ∆ is an open subset of R² and X is a vector field of class C r with r ≥ 1. For the basic definition ofω-limit set or α-limit set of an orbit, see for instance Chapter 1 of (Dumortier et al. 2006).

Theorem 2 (Poincaré-Bendixson Theorem). Let φ(t) = φ(t, p) be an integral curve of X defined for all t≥ 0, such that φ(0) = p and = {φ(t) : t ≥ 0} is contained in a compact set K ⊂ ∆. Assume that the vector fieldX has at most a finite number of singularities in K. Then one of the following statements holds.

1. (i) If ω(p) contains only regular points, then ω(p) is a periodic orbit.

2. (ii) If ω(p) contains both regular and singular points, then ω(p) is a graphic, i.e. a set formed by orbits, each of them tending to one of the singular points contained in ω(p) as t → ± ∞.

3. (iii) If ω(p) does not contain regular points, then ω(p) is a unique singular point.

The Poincaré-Bendixson theorem can also be stated for α-limit sets. The next result is the Poincaré-Hopf theorem for the Poincaré compactification of a polynomial vector field. For a proof, see Theorem 6.30 of (Dumortier et al. 2006).

Theorem 3.Let X be a polynomial vector field. If p(X )defined on the Poincaré sphere S² has finitely many singular points, then the sum of their topological indices is two

The next result of Sabatini, see Theorem 2.3 of (Sabatini 1998), will play a main role in the proof of Theorem 1.

Theorem 4.Let F = (f, g) be a polynomial map with nonvanishing Jacobian determinant such that F (0, 0) = (0, 0). Then the following properties are equivalent

1. (1) The origin is a global center for the polynomial differential system = −f fy −ggy , = f fx +ggx.

2. (2) F is a global diffeomorphism of the plane onto itself.

3- PROOF OF THEOREM 1

We denote (a1, a2) = F (0,0) and consider the translation A(x, y) = (x − a1, y −a2). Taking the map G = A ◦F, we observe that G(0, 0) = (0, 0), det DG is nowhere zero, the degrees of the components of G are equal and the assumption of Theorem 1 is still true for G, because the higher degree terms of Fand G coincide. Moreover, F is injective if and only if G is injective. In what follows we will assumeF = G.

We consider now the function HF : R² → R defined byHF (x, y) = (f (x, y)² + g(x, y)²⁾ /2 and the Hamiltonian vector field associated toHF , X = (P, Q), i.e

It is clear that if (x0, y0) is such that F(x0, y0) = (0, 0), then (x0, y0) is a singular point of X. Moreover, (x0, y0) is an isolated minimum ofHF and so it is a center of the Hamiltonian system (1) because near (x0, y0) the level curves ofHF are closed.

By Theorem 4, in order to prove Theorem 1 it is enough to prove that (0,0) is a global center of the polynomial differential system (1).

We now consider the Poincaré compactification p(X ) of X defined in S².

We claim that p(X ) does not have infinite singular points.Indeed, there exist singular points of p(X ) at infinity if and only if there exist linear factors dividing the homogeneous polynomialyPn − xQn, where n is the maximum degree of P and Q. Let mbe the degree of the polynomials f and g. It is clear that n ≤ 2m − 1. Moreover, by the Euler's Theorem for homogeneous functions it follows that

and so the homogeneous part of degree 2m − 1 of Por Q is not zero, proving n = 2m− 1. The same calculation (2) also shows that a linear factor dividesyPn − xQn if and only if it dividesfm = and gm = , which does not occur by assumption. So the claim is proved and it follows that S¹ is a periodic orbit of p(X ).

We claim that system (1) has no finite singular points, but the origin.

Indeed, P (x, y) = Q(x, y) = 0 is equivalent to

which gives that f (x, y) =g(x, y) = 0, since det DF(x, y) ≠ 0. Thus all the finite singular points of X are zeros of F , and consequently there are just a finite number of them. Moreover, by the considerations on the functionHF above, all the finite singular points of X are centers, each of them producing two centers of p(X ) in S² (one in H+ and one in H−). As there are no infinite singular points, it follows by Theorem 3 that the sum of the indices of the singular points of p(X ) is 2. Since each center has index 1, it follows that p(X ) has only two singular points, and thus X has only one singular point. This singular point is (0, 0). Hence, the claim is proven.

Now we shall prove that (0, 0) is a global center ofp(X ). Then (0, 0) will be a global center of X, and by Theorem 4 the proof of Theorem 1 will be finished.

From now on we will consider p(X ) the projection of the Poincaré compactification on D2. Since there are no finite singular points, except the origin, and there are no infinite singular points, the boundary of the period annulus P of the center (0, 0) is a periodic orbit that we call °c. If it is S¹, we are done. If not, in a neighborhood of °c in the exterior of the period annulus we take the orbit °a through some point a in this region, and we claim that °a has its ω-limit or its α-limit set equal to °c. Indeed by the Poincaré-Bendixson Theorem (see Theorem 2), these ω- or α-limit sets are either a singular point, a graphic, or a periodic orbit. Since in D2 \P there are no singular points of p(X ), such ω- andα-limit sets are periodic orbits. This implies that °c outside P is stable or unstable, i.e. the orbits near it outside P spiral in forward or backward time to it. This completes the proof of the claim.

Considering now the Poincaré map defined in a transversal section Sthrough °c, we observe that it is the identity map in S ∩ P, and it is different from the identity in S ∩ (D2 \ P). But this is impossible, because the Poincaré map is an analytic function since the vector field p(X ) is analytic. Therefore, the center is global and this completes the proof of Theorem 1.

Remark 5.Analyzing the proof of Theorem 1, it is tempting to think that it can also be done under the hypothesis degree of f greater than degree of g. The hypothesis (2) in such a version of Theorem 1 would be that there are no real linear factors dividing . The problem is that this assumption guarantees that f is not a submersion, a necessarycondition for det DF to be nowhere zero. Indeed, if f is a submersion, then the vector field Y = (fy, − fx) has no finite singular points. Since the Poincaré compactification of Y defined inS² , an even dimensional sphere, is a smooth vector field, by the Poincaré-Hopf Theorem (see for instance Theorem 6.30 of ( Dumortier et al. 2006 )) it must have a singular point, which will be an infinite one. Then a real linear factor divides , where m is the degree of f .

4- PINCHUK COUNTEREXAMPLE

The above mentioned Pinchuk example is F (x, y) = (p(x, y), q(x, y)) with p and q as follows. Let t = xy − 1, h = t(xt + 1), f = (xt + 1)2(t2 + y) and define p(x, y) = h + f . Observe that p has degree 10. The polynomial q(x, y) varies for different Pinchuk maps, but always has the form q(x, y) = −t 2 − 6th(h+1) − u(f, h), where u is an auxiliary polynomial in f and h, chosen so that the Jacobian determinant of (p, q) is t 2 + (t + f (13 + 15h))2 + f 2, which is strictly positive. This u(f, h) is a solution of a differential equation.

In the original paper (Pinchuk 1994) it is suggested the following u:

which gives q of degree 40, since the higher degree of qcomes from the term with f 4 contained inu. So the map (p, q) does not satisfy the hypothesis of Theorem 1 that the degrees of p and qmust be equal.

In (Campbell 2011) the polynomial uis taken

u = 170f h + 91h ² + 195f h ² + 69h ³ + 75f h ³ + 75h ⁴/4,

which gives q of degree 25, since the higher degree comes from the term with 75f h3 contained in u. So the degrees ofp and q are also not equal.

We remark that 25 is the smallest degree that a component q in a Pinchuk map can have. For details see (Campbell 2011).

ACKNOWLEDGMENTS

The second author is partially supported by a Ministerio de Economia y Competitividad (MINECO) grant MTM2013-40998-P, an Agència de Gestió d'Ajuts Universitaris i de Recerca (AGAUR) grant number 2014SGR-568, the grants from European Commission (FP7-PEOPLE-2012-IRSES) 318999 and 316338. The two authors are also supported by a Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) grant number 88881.030454/2013-01 from the program Ciência sem Fronteiras - Professor Visitante Especial (CSF-PVE).

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• AMS(2010): 14R15, 35F05, 35A30

Publication Dates

• Publication in this collection
  25 Aug 2015
• Date of issue
  Sept 2015

History

• Received
  04 Nov 2013
• Accepted
  23 Jan 2015