Abstract
Abstract: We present two new classes of polynomial maps satisfying the real Jacobian conjecture in . The first class is formed by the polynomials maps of the form (q(x)–p(y), q(y)+p(x)) : such that p and q are real polynomials satisfying p'(x)q'(x) ≠ 0. The second class is formed by polynomials maps (f, g): where f and g are real homogeneous polynomials of the same arbitrary degree satisfying some conditions.
Key words
injective polynomial maps; global center; real Jacobian conjecture; planar Hamiltonian systems
Introduction
Let be a polynomial map such that its Jacobian never vanishes. The celebrated real Jacobian conjecture states that under these conditions is injective. This conjecture goes back to 1939, see Keller (1939)KELLER OH. 1939. Ganze Cremona-Transformationen. Monatsh Math Phys 47(1): 299–306. .
In 1994 Pinchuk (1994)PINCHUK S. 1994. A counterexample to the strong real Jacobian conjecture. Math Z 217(1): 1–4. found a map with and polynomials of degree 10 and 25 respectively, and with Jacobian strictly positive, such that is not injective.
Although the real Jacobian conjecture has been proved false by Pinchuk, a considerable number of papers has been devoted to this subject, mainly searching for additional conditions such that the conjecture might hold. The problem of determining ifis injective in the case of its Jacobian to be a non–zero constant, known as the Jacobian conjecture, is still open, see Essen (2000)ESSEN AVD. 2000. Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics 190. Basel: Birkhauser Verlag. and the references therein for more information.
Main Results
In this note we present two new classes of polynomial maps that satisfies the real Jacobian conjecture. In what follows we present our main results.
Theorem 1. Letand be real polynomials of one variable and consider the polynomial mapwith ,and . Then the Jacobian of polynomial map never vanish and is injective.
Theorem 1. is proved in the section Proofs of the Theorems.
As usual here denotes the derivative of with respect to the variable , and if we denote by the partial derivative of with respect to the variable . Similarly is defined .
Theorem 2. Letand be real homogeneous polynomials of the same degree in the variablesand such that
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the Jacobian of the polynomial mapnever vanish,
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and have no real linear factors in common,
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the polynomials and have no real common factors, and
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Then the polynomial map is injective.
Theorem 2 is also proved in the section Proofs of the Theorems.
Other classes of polynomial maps satisfying the real Jacobian conjecture were given in Braun and Llibre (2015)BRAUN F AND LLIBRE J. 2015. A new qualitative proof of a result on the real jacobian conjecture. An Acad Bras Cienc 87: 1519–1524. and Braun, Giné, and Llibre (2016).BRAUN F, GINÉ J AND LLIBRE J. 2016. A sufficient condition in order that the real Jacobian conjecture in R2 holds. J Differential Equations 260(6): 5250–5258.
Preliminary results
Let and be polynomials in the variables and . Consider the polynomial differential system
We say that an isolated singularity of system (1) is a center when there is a neighborhoodof such that every solution in is periodic. The biggest open connected set containing, denoted by , such that is filled with periodic orbits is called the period annulus of the center. If then is a global center of system(1) .
Let be an isolated singularity of system(1) . Ifthen is the Jacobian matrix of system(1) at . If then we say that is a non–degenerate singular point. Under these assumptions a necessary condition in order that be a center is that the eigenvalues of are purely imaginary. Such a center is a non-degenerate center.
The polynomial differential system(1) is a Hamiltonian system if there is a polynomial such that and . Then the polynomial is called the Hamiltonian of the Hamiltonian system(1) .
The following result is due to Sabatini, see Theorem 2.3 of Sabatini (1998)SABATINI M. 1998. A connection between ishochronous Hamiltonian centres and the Jacobian Conjecture. Nonlinear Anal 34: 829–838. . This result provides a relation between the real Jacobian conjecture and the global centers of some polynomial Hamiltonian systems.
Theorem 3. Let be a polynomial map such that its Jacobian never vanishes and . Then the following statements are equivalent.
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The polynomial Hamiltonian system with Hamiltonian has a global center at the origin of coordinates.
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The map is a global diffeomorphism of onto itself.
In addition we present the following result due to Braun and Llibre, see Lemma 1 of Braun and Llibre (2017)BRAUN F AND LLIBRE J. 2017. On the connection between global centers and global injectivity. Preprint.
Lemma 4. Letbe a function defined in an open connected setand such that the Jacobian of in does not vanish. Moreover consider the Hamiltonian . Then is a singular point of the polynomial Hamiltonian system if and only if . Under these conditions is a non-degenerate center and also an isolated global minimum of . In particular if the Jacobian of never vanishes in all the singular points of the Hamiltonian system,in are non–degenerate centers and are the zeros of the map.
The Poincaré compactification
The set is called the Poincaré sphere. Consider the tangent space to the Poincaré sphere at the pointand the central projection .
Let be a polynomial vector field of degree in the plane . The map defines 2 copies of in , one in the northern hemisphere and the other in the southern hemisphere. Let be the vector field defined on the Poincaré sphere except on its equator. We remark that is everywhere tangent to and is identified to the infinity of . We define , the Poincaré compactified vector field associated toas the analytic extension of to Note that studying the behavior of around , we obtain the behavior of at infinity. Also, is invariant under the flow of .
The Poincaré disc is the projection of the closed northern hemisphere of on under .
The singular points of in the interior of the Poincaré disc, or equivalently in open northern hemisphere , are called the finite singular points of . While the singular points of contained in are called the infinite singular points of .
For more details on the Poincaré compactification, see chapter 5 of Dumortier, Llibre, and Artés (2006)DUMORTIER F, LLIBRE J AND Artés J. 2006. Qualitative theory of planar differential systems. Berlin Heidelberg: Spring-Verlag. .
The following result is the Poincaré–Hopf Theorem for the Poincaré compactification of a polynomial vector field. For a proof see for instance Theorem 6.30 of Dumortier, Llibre, and Artés (2006).
Theorem 5. Letbe a polynomial vector field. If defined on the Poincaré sphere has finitely many singular points, then the sum of their topological indices is two.
Proofs of the theorems
Proof of Theorem 1.We claim that the Jacobian of is . Indeed, since each one of the one-variable polynomials,is either strictly positive or strictly negative. Consider the case and , the other cases can be done similarly. Then . But whatever the signals of each of these polynomials are, we have that and , consequently , and the claim is proved.
From the claim we have that the polynomial map with ,satisfies the assumptions of Lemma 4 with. Hence all the singular points of the polynomial Hamiltonian system with Hamiltonian are non–degenerated centers.
We study the infinite singular points of the system
where ,with ,,and positive integers, , and l.o.t. means lower order terms of the polynomial.
First we assume that . For studying the infinite equilibria consider the homogeneous polynomial of degree , where and are the homogeneous parts of degree of the polynomials and respectively.
Since for , the Hamiltonian system (2) has no infinite singular points.
For the cases and we have respectively and for . Therefore again the Hamiltonian system has no infinite singular points, and is a periodic orbit of the Poincaré compactification of system (2).
In summary we know that all the finite singular points of system(2) are centers, and that it has no infinite singular points. Moreover, since this system is polynomial it has finitely many singular points, therefore by the Poincaré–Hopf Theorem (Theorem 5) applied to the Poincaré sphere, we obtain that two times the sum of the indices of the finite singular points is equal to . One of the “two times” comes from the northern open hemisphere and the other from the south open hemisphere of the Poincaré sphere because we have a copy in each of these hemispheres of our system(2). Hence since the sum of the indices of all the finite singular points is, and each center has index , the system has a unique center, which we denote by.
To end the proof applying Theorem 3 we must prove that the local center is global. But for applying Theorem 3 we need that, so we consider the map , where if . Then the map satisfies and also satisfies all the conditions of the mapgiven in the statement of Theorem 1. So we shall prove thatis injective, and consequently will be injective. In what follows for simplifying the notation we denote by .
Let be the period annulus of . If the last periodic orbit of is the infinity in the Poincaré disc we are done. Assume thatis the last periodic orbit of and that it does not coincide with the periodic orbit at, and let be a point of . Consider the Poincaré map associated to , where is a local transverse section to the vector field associated to system(2) through the point , for further information on these topics see chapter 1 of Dumortier, Llibre, and Artés (2006). Here denotes the domain of definition of the mapon the section .
By Proposition 1.21 of Dumortier, Llibre, and Artés (2006) the mapis analytic because system(2) is polynomial. Clearly the map restricted to the part of contained in the period annulus is the identity. Therefore, since is an analytic map of one variable, it is analytic in the whole. Hence cannot be the last periodic orbit of , a contradiction. Consequently the center is global and by Theorem 3 we conclude that is injective.
Proposition 4.2 of Cima and Llibre (1990)CIMA A AND LLIBRE J. 1990. Algebraic and topological classification of the homogeneous cubic vector fields in the plane. J Math Anal Appl 147(2): 420–448. states:
Proposition 6. Letand be two real homogeneous polynomials of degreein the variables and . Assume that and do not have real common factors, that has no real linear factors, and that
Then the phase portrait of the polynomial vector fieldis a global center.
Proof of Theorem 2. Under the assumptions of Theorem 2, first we shall see that the polynomial has no real linear factors. By the Euler’s Theorem for homogeneous functions we have that
Therefore since the homogeneous polynomialsand has no real linear factors in common, the homogeneous polynomial also does not have a real linear factor. Hence, from the hypotheses of Theorem 2 all the assumptions of Proposition 6 are satisfied. Consequently the Hamiltonian system with Hamiltonianhas a global center. So by Theorem 3 we get that the polynomial map is injective. We note that as in the proof of Theorem 1, for applying Theorem 3 we need that , again taking the map , where if , we verify that and that satisfies the conditions of the map given in the statement of Theorem 2. So we apply Theorem 3 to the map , and we obtain the injectivity for the map, and consequently for the map.
ACKNOWLEGMENTS
We thank to the reviewer his/her comments which help us to improve the presentation of our results.The first author is supported by a Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) grant number 2016/23285-9. The second author is supported by a MINECO-FEDER grant MTM2016-77278-P, a MINECO grant MTM2013-40998-P, and an AGAUR grant number 2014SGR-568.
- BRAUN F, GINÉ J AND LLIBRE J. 2016. A sufficient condition in order that the real Jacobian conjecture in R2 holds. J Differential Equations 260(6): 5250–5258.
- BRAUN F AND LLIBRE J. 2015. A new qualitative proof of a result on the real jacobian conjecture. An Acad Bras Cienc 87: 1519–1524.
- BRAUN F AND LLIBRE J. 2017. On the connection between global centers and global injectivity. Preprint
- CIMA A AND LLIBRE J. 1990. Algebraic and topological classification of the homogeneous cubic vector fields in the plane. J Math Anal Appl 147(2): 420–448.
- DUMORTIER F, LLIBRE J AND Artés J. 2006. Qualitative theory of planar differential systems. Berlin Heidelberg: Spring-Verlag.
- ESSEN AVD. 2000. Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics 190. Basel: Birkhauser Verlag.
- KELLER OH. 1939. Ganze Cremona-Transformationen. Monatsh Math Phys 47(1): 299–306.
- PINCHUK S. 1994. A counterexample to the strong real Jacobian conjecture. Math Z 217(1): 1–4.
- SABATINI M. 1998. A connection between ishochronous Hamiltonian centres and the Jacobian Conjecture. Nonlinear Anal 34: 829–838.
Publication Dates
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Publication in this collection
01 July 2019 -
Date of issue
2019
History
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Received
14 Sept 2017 -
Accepted
23 Aug 2018