The Nagata Type Polynomial Automorphisms and Rectifiable Space Lines

This note relates polynomial remainders with polynomial automorphisms of the plane. It also formulates a conjecture, equivalent to the famous Jacobian Conjecture. The latter provides an algorithm for checking when a polynomial map is an automorphism. In addition, a criterion is presented for a real polynomial map to be bijective.

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Polynomial Automorphisms and Gröbner Reductions

Journal of Algebra ◽

10.1006/jabr.1997.7083 ◽

1997 ◽

Vol 197 (2) ◽

pp. 546-558 ◽

Cited By ~ 17

Author(s):

Vladimir Shpilrain ◽

Jie-Tai Yu

Keyword(s):

Polynomial Automorphisms

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Hypercyclic operators on algebra of symmetric analytic functions on $\ell_p$

Carpathian Mathematical Publications ◽

10.15330/cmp.8.1.127-133 ◽

2016 ◽

Vol 8 (1) ◽

pp. 127-133 ◽

Cited By ~ 1

Author(s):

Z.G. Mozhyrovska

Keyword(s):

Analytic Functions ◽

Entire Functions ◽

Composition Operators ◽

Translation Operator ◽

Hypercyclic Operators ◽

Polynomial Automorphisms ◽

Fréchet Algebra

In the paper, it is proposed a method of construction of hypercyclic composition operators on $H(\mathbb{C}^n)$ using polynomial automorphisms of $\mathbb{C}^n$ and symmetric analytic functions on $\ell_p.$ In particular, we show that an "symmetric translation" operator is hypercyclic on a Frechet algebra of symmetric entire functions on $\ell_p$ which are bounded on bounded subsets.

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Polynomial Automorphisms of Soluble Groups

Communications in Algebra ◽

10.1080/00927870802502837 ◽

2009 ◽

Vol 37 (10) ◽

pp. 3388-3400

Author(s):

Gérard Endimioni

Keyword(s):

Soluble Groups ◽

Polynomial Automorphisms

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Co-tame polynomial automorphisms

International Journal of Algebra and Computation ◽

10.1142/s0218196719500292 ◽

2019 ◽

Vol 29 (05) ◽

pp. 803-825 ◽

Cited By ~ 1

Author(s):

Eric Edo ◽

Drew Lewis

Keyword(s):

Characteristic Zero ◽

Independent Interest ◽

Technical Tool ◽

Polynomial Automorphism ◽

Polynomial Automorphisms ◽

Negative Results ◽

Dimension Formula ◽

Positive Results ◽

Main Technical Tool

A polynomial automorphism of [Formula: see text] over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms of [Formula: see text], including nonaffine [Formula: see text]-triangular automorphisms, are co-tame. Of particular interest, if [Formula: see text], we show that the statement “Every [Formula: see text]-triangular automorphism is either affine or co-tame” is true if and only if [Formula: see text]; this improves upon positive results of Bodnarchuk (for [Formula: see text], in any dimension [Formula: see text]) and negative results of the authors (for [Formula: see text], [Formula: see text]). The main technical tool we introduce is a class of maps we term translation degenerate automorphisms; we show that all of these are either affine or co-tame, a result that may be of independent interest in the further study of co-tame automorphisms.

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