A characterization of Keller maps

Jędrzejewicz, Piotr

doi:10.1016/j.jpaa.2012.06.015

Cited by 7 publications

(7 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following lemma is a natural generalization of [12], Lemma 3.1. For the proof it is enough to add the argument from the beginning of the proof of [13], Proposition 3.4.a with Q = (g).…”

Section: Analogs Of Jacobian Conditions In Terms Of Irreducible and Smentioning

confidence: 99%

“…(i) ⇒ (ii) We combine the arguments from proofs of [12], Theorem 4.1, (i) ⇒ (ii) and [13], Theorem 3.6 (⇒). Assume that g | dgcd(f 1 , .…”

Section: Analogs Of Jacobian Conditions In Terms Of Irreducible and Smentioning

confidence: 99%

“…For more information on the Jacobian Conjecture we refer the reader to van den Essen's book [7]. One of the authors obtained in [12] a characterization of endomorphisms satisfying the Jacobian condition (2), where k is a field of characteristic zero, as mapping irreducible polynomials to square-free ones. De Bondt and Yan proved in [4] that mapping square-free polynomials to square-free ones is also equivalent to (2).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analogs of Jacobian conditions for subrings

Jędrzejewicz

Zieliński

2017

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

We present a generalization of the Jacobian Conjecture for m polynomials in n variables:, where k is a field of characteristic zero and m ∈ {1, . . . , n}. We express the generalized Jacobian condition in terms of irreducible and square-free elements of the subalgebra k[f 1 , . . . , f m ]. We also discuss obtained properties in a more general setting -for subrings of unique factorization domains.

show abstract

“…The following lemma is a natural generalization of [12], Lemma 3.1. For the proof it is enough to add the argument from the beginning of the proof of [13], Proposition 3.4.a with Q = (g).…”

Section: Analogs Of Jacobian Conditions In Terms Of Irreducible and Smentioning

confidence: 99%

“…(i) ⇒ (ii) We combine the arguments from proofs of [12], Theorem 4.1, (i) ⇒ (ii) and [13], Theorem 3.6 (⇒). Assume that g | dgcd(f 1 , .…”

Section: Analogs Of Jacobian Conditions In Terms Of Irreducible and Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Analogs of Jacobian conditions for subrings

Jędrzejewicz

Zieliński

2017

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…Let us remark that the zero characteristic analog of Theorem 4.8 for m = n ( [25], Theorem 4.1) is connected with a characterization of Keller maps and an equivalent formulation of the Jacobian Conjecture.…”

Section: Example 43 Consider Polynomialsmentioning

confidence: 99%

Rings of constants of polynomial derivations and p-bases

Jędrzejewicz¹

2013

Analytic and Algebraic Geometry

View full text Add to dashboard Cite

Abstract. We present a survey of results concerning p-bases of rings of constants with respect to polynomial derivations in characteristic p > 0. We discuss characterizations of rings of constants, properties of their generators and a general characterization of their p-bases. We also focus on some special cases: one-element p-bases, eigenvector p-bases and when a ring of constants is a polynomial graded subalgebra. IntroductionIn Section 1 we introduce the notation and definitions concerning derivations, rings of constants and p-bases. Then we discuss characterizations of rings of constants in Section 2 and we present some basic information on the number of generators for rings of constants of polynomial derivations in Section 3. For a wider panorama of contemporary differential algebra we refer to the book of Nowicki ([41]), and for problems connected with locally nilpotent derivations we refer to the book of Freudenburg ([10]).Next two sections contain a general characterization of p-bases of rings of constants with respect to polynomial derivations, based on the author's paper [26]. In Section 4 we present generalizations of Freudenburg's lemma (Theorems 4.7 and 4.8). The main theorem (Theorem 5.4) and its motivations are presented in Section 5. In Section 6 (based on the results of [23] and [18]) we discuss analogies and differences between single generators of rings of constants in zero and positive characteristic, and we focus on some special cases. Section 7, based on [24], is devoted to specific properties of eigenvector p-bases (Theorem 7.2). Finally, in 2010 Mathematics Subject Classification. Primary 13N15, Secondary 13F20.

show abstract

“…Namely, we show that in the case of the polynomial algebra, the condition (i) is also necessary, and we strengthen the condition (iv) to make it also sufficient. The crucial fact we need, to realize this aim, is the positive characteristic version of Freudenburg's lemma for polynomials, where the zero characteristic version was obtained in [11,Theorem 4.1]. Note also that a characterization of one-element -bases was obtained in [8,Theorem 4.2].…”

Section: Introductionmentioning

confidence: 99%

A characterization of p-bases of rings of constants

Jędrzejewicz

2013

Open Mathematics

Self Cite

View full text Add to dashboard Cite

We obtain two equivalent conditions for polynomials in variables to form a -basis of a ring of constants of some polynomial K -derivation, where K is a unique factorization domain of characteristic > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case = this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic. MSC:13N15, 13F20, 14R15

show abstract

A characterization of Keller maps

Cited by 7 publications

References 12 publications

Analogs of Jacobian conditions for subrings

Analogs of Jacobian conditions for subrings

Rings of constants of polynomial derivations and p-bases

A characterization of p-bases of rings of constants

Contact Info

Product

Resources

About