Displaying similar documents to “Polynomial Automorphisms Over Finite Fields”

Polynomial automorphisms over finite fields: Mimicking tame maps by the Derksen group

Maubach, Stefan, Willems, Roel (2011)

Serdica Mathematical Journal

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2010 Mathematics Subject Classification: 14L99, 14R10, 20B27. If F is a polynomial automorphism over a finite field Fq in dimension n, then it induces a permutation pqr(F) of (Fqr)n for every r О N*. We say that F can be “mimicked” by elements of a certain group of automorphisms G if there are gr О G such that pqr(gr) = pqr(F). Derksen’s theorem in characteristic zero states that the tame automorphisms in dimension n і 3 are generated by the affine maps and the one map (x1+x22,...

The solution of the Tame Generators Conjecture according to Shestakov and Umirbaev

Arno van den Essen (2004)

Colloquium Mathematicae

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The tame generators problem asked if every invertible polynomial map is tame, i.e. a finite composition of so-called elementary maps. Recently in [8] it was shown that the classical Nagata automorphism in dimension 3 is not tame. The proof is long and very technical. The aim of this paper is to present the main ideas of that proof.

The Automorphism Group of the Free Algebra of Rank Two

Cohn, P. (2002)

Serdica Mathematical Journal

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The theorem of Czerniakiewicz and Makar-Limanov, that all the automorphisms of a free algebra of rank two are tame is proved here by showing that the group of these automorphisms is the free product of two groups (amalgamating their intersection), the group of all affine automorphisms and the group of all triangular automorphisms. The method consists in finding a bipolar structure. As a consequence every finite subgroup of automorphisms (in characteristic zero) is shown to be conjugate...

Tame Automorphisms of ℂ³ with Multidegree of the Form (p₁,p₂,d₃)

Marek Karaś (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let d₃ ≥ p₂ > p₁ ≥ 3 be integers such that p₁,p₂ are prime numbers. We show that the sequence (p₁,p₂,d₃) is the multidegree of some tame automorphism of ℂ³ if and only if d₃ ∈ p₁ℕ + p₂ℕ, i.e. if and only if d₃ is a linear combination of p₁ and p₂ with coefficients in ℕ.

On reconstruction of polynomial automorphisms

Paweł Gniadek (1996)

Annales Polonici Mathematici

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We extend results on reconstructing a polynomial automorphism from its restriction to the coordinate hyperplanes to some wider class of algebraic surfaces. We show that the algorithm proposed by M. Kwieciński in [K2] and based on Gröbner bases works also for this class of surfaces.

Wild Multidegrees of the Form (d,d₂,d₃) for Fixed d ≥ 3

Marek Karaś, Jakub Zygadło (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let d be any integer greater than or equal to 3. We show that the intersection of the set mdeg(Aut(ℂ³))∖ mdeg(Tame(ℂ³)) with {(d₁,d₂,d₃) ∈ (ℕ ₊)³: d = d₁ ≤ d₂≤ d₃} has infinitely many elements, where mdeg h = (deg h₁,...,deg hₙ) denotes the multidegree of a polynomial mapping h = (h₁,...,hₙ): ℂⁿ → ℂⁿ. In other words, we show that there are infinitely many wild multidegrees of the form (d,d₂,d₃), with fixed d ≥ 3 and d ≤ d₂ ≤ d₃, where a sequence (d₁,...,dₙ)∈ ℕ ⁿ is a wild multidegree...

Linear differential equations and Hurwitz series

William F. Keigher, V. Ravi Srinivasan (2011)

Banach Center Publications

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In this article, we study solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of the solutions. Moreover, we give explicit formulas that compute the group of differential automorphisms. We require neither that the underlying field be algebraically closed nor that the characteristic of the field be zero.

A note on central automorphisms of groups

Giovanni Cutolo (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A characterization of central automorphisms of groups is given. As an application, we obtain a new proof of the centrality of power automorphisms.