We first show that if there is a counterexample to the two-dimensional Jacobian conjecture,then there exists a nonempty closed smooth affine surface in C^(2)×C^(2)satisfying various properties.Next,we conjecture ...We first show that if there is a counterexample to the two-dimensional Jacobian conjecture,then there exists a nonempty closed smooth affine surface in C^(2)×C^(2)satisfying various properties.Next,we conjecture that such a surface does not exist.展开更多
In this paper, we introduce a polynomial sequence in K[x], in which two neighbor polynomials satisfy a wonderful property. Using that,we give partial answer of an open problem： ifφ（x, y, z） = （f（x, y）, g（x, y...In this paper, we introduce a polynomial sequence in K[x], in which two neighbor polynomials satisfy a wonderful property. Using that,we give partial answer of an open problem： ifφ（x, y, z） = （f（x, y）, g（x, y, z）, z）, which sends every linear coordinate to a coordinate, then φ is an automorphism of K[x, y, z]. As a byproduct, we give an easy proof of the well-known Jung＇s Theorem.展开更多
We classify all polynomial maps of the form H=(u(x,y,z),v(x,y,z),h(x,y))in the case when the Jacobian matrix of H is nilpotent and the highest degree of z in v is no more than 1.In addition,we generalize the structure...We classify all polynomial maps of the form H=(u(x,y,z),v(x,y,z),h(x,y))in the case when the Jacobian matrix of H is nilpotent and the highest degree of z in v is no more than 1.In addition,we generalize the structure of polynomial maps H to H=(H_(1)(x_(1),x_(2),…,x_(n)),b_(3)x_(3)+…+b_(n)x_(n)+H^((0))_(2)(x_(2)),H_(3)(x_(1),x_(2)),…,H_(n)(x_(1),x_(2))).展开更多
Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showe...Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducibility properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix （over C and hence any field of characteristic zero） are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called （planar） weak Jacobian Conjecture by Kaliman.展开更多
Two characterizations for a local diffeomorphism of R^n to be global one aregiven in terms of associated Wazewski equations. The two characterizations could be useful for theinvestigation of the Jacobian conjecture.
文摘We first show that if there is a counterexample to the two-dimensional Jacobian conjecture,then there exists a nonempty closed smooth affine surface in C^(2)×C^(2)satisfying various properties.Next,we conjecture that such a surface does not exist.
文摘In this paper, we introduce a polynomial sequence in K[x], in which two neighbor polynomials satisfy a wonderful property. Using that,we give partial answer of an open problem： ifφ（x, y, z） = （f（x, y）, g（x, y, z）, z）, which sends every linear coordinate to a coordinate, then φ is an automorphism of K[x, y, z]. As a byproduct, we give an easy proof of the well-known Jung＇s Theorem.
基金Supported by the National Natural Science Foundation of China(Grant No.11601146,11871241)the Natural Science Foundation of Hunan Province(Grant No.2016JJ3085)the Construct Program of the Key Discipline in Hunan Province.
文摘We classify all polynomial maps of the form H=(u(x,y,z),v(x,y,z),h(x,y))in the case when the Jacobian matrix of H is nilpotent and the highest degree of z in v is no more than 1.In addition,we generalize the structure of polynomial maps H to H=(H_(1)(x_(1),x_(2),…,x_(n)),b_(3)x_(3)+…+b_(n)x_(n)+H^((0))_(2)(x_(2)),H_(3)(x_(1),x_(2)),…,H_(n)(x_(1),x_(2))).
基金The first author was supported by the Netherlands Organisation for Scientific Research （NWO）. The second author was supported by the National Natural Science Foundation of China （Grant No. 11371343）.
文摘Jedrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducibility properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix （over C and hence any field of characteristic zero） are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called （planar） weak Jacobian Conjecture by Kaliman.
文摘Two characterizations for a local diffeomorphism of R^n to be global one aregiven in terms of associated Wazewski equations. The two characterizations could be useful for theinvestigation of the Jacobian conjecture.
