P -Conjecture for tame automorphisms of C3

The present paper deals with permutations induced by tame automorphisms over finite fields. The first main result is a formula for determining the sign of the permutation induced by a given elementary automorphism over a finite field. The second main result is a formula for determining the sign of the permutation induced by a given affine automorphism over a finite field. We also give a combining method of the above two formulae to determine the sign of the permutation induced by a given triangular automorphism over a finite field. As a result, for a given tame automorphism over a finite field, if we know a decomposition of the tame automorphism into a finite number of affine automorphisms and elementary automorphisms, then one can easily determine the sign of the permutation induced by the tame automorphism.

Let $F=(F_{1},\ldots ,F_{n}):\mathbb {C}^{n}\rightarrow \mathbb {C}^{n}$ be any polynomial mapping. The multidegree of $F$, denoted $\textrm {mdeg} F,$ is the sequence of positive integers $(\deg F_{1},\ldots ,\deg F_{n}).$ In this paper we address the following problem: for which sequence $(d_{1},\ldots ,d_{n})$ is there an automorphism or a tame automorphism $F:\mathbb {C}^{n}\rightarrow \mathbb {C}^{n}$ with $\textrm {mdeg} F=(d_{1},\ldots ,d_{n})$? We prove, among other things, that there is no tame automorphism $F:\mathbb {C}^{3}\rightarrow \mathbb {C}^{3}$ with $\textrm {mdeg} F=(3,4,5)$.

To an endomorphism ø of C[xl,…, xn] we associate co(ø) the dimension of the smallest sub-coalgebra C(ø) of C[G n a] containing the ø(xi). If the Jacobian of ø is invertible, C(ø) will be a generating set. If co(ø) is minimal (that is n +1), then ø is a tame automorphism. We use Lie stacks to construct such tame automorphisms and link their study to the classification problem of local cornmutative Galgebras of dimension n +1.

Let $3\leq d_1\leq d_2\leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $\frac{d_1}{\gcd(d_1,d_3)}\neq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3\in d_1\mathbb{N}+d_2\mathbb{N}$; (2) If $d_3$ is a prime number and $\gcd(d_1,d_2)=1$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_3\in d_1\mathbb{N}+d_2\mathbb{N}$. We also relate this investigation with a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example to this conjecture.

Let [Formula: see text] be a field of characteristic zero. For positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], let [Formula: see text] be a free center-by-metabelian and nilpotent Lie algebra over [Formula: see text] of rank [Formula: see text] and class [Formula: see text], freely generated by a set [Formula: see text]. It is shown that the automorphism group [Formula: see text] of [Formula: see text] is generated by the general linear group [Formula: see text] and two more IA-automorphisms. Let [Formula: see text] be the field of rational numbers. We give [Formula: see text] the structure of a group, say [Formula: see text], via the Baker–Campbell–Hausdorff formula. Let [Formula: see text] be the subgroup of [Formula: see text] generated by [Formula: see text]. We prove that the subgroup of [Formula: see text] generated by the tame automorphisms [Formula: see text] and three more IA-automorphisms of [Formula: see text] has finite index in [Formula: see text]. For [Formula: see text], the subgroup of [Formula: see text] generated by the tame automorphisms [Formula: see text] and two more IA-automorphisms of [Formula: see text] has finite index in [Formula: see text]. A similar result is proved for the automorphism group of a free center-by-metabelian and nilpotent group of rank [Formula: see text] and class [Formula: see text].

We study the group Tame(SL 2 ) of tame automorphisms of a smooth affine 3-dimensional quadric, which we can view as the underlying variety of SL 2 (ℂ). We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is CAT(0) and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in Tame(SL 2 ) is linearizable, and that Tame(SL 2 ) satisfies the Tits alternative.

Acylindrical hyperbolicity of the three-dimensional tame automorphism group

Denote a free Leibniz algebra in two variables over a field K by LB ⟨ x, y ⟩. The groups of almost triangular AT(LB ⟨ x, y ⟩) and almost tame automorphisms Tame(LB ⟨ x, y ⟩) of LB ⟨ x, y ⟩ are introduced here. These groups extend the groups of triangular and tame automorphisms, respectively. We prove that where H = GL 2(K) ∩ AT(LB ⟨ x, y ⟩). We also show that Tame(LB ⟨ x, y ⟩) is smaller than the group of automorphisms of LB ⟨ x, y ⟩.

This paper surveys results concerning the quantization approach to the Jacobian Conjecture and related topics on noncommutative algebras. We start with a brief review of the paper and its motivations. The first section deals with the approximation by tame automorphisms and the Belov–Kontsevich Conjecture. The second section provides quantization proof of Bergman’s centralizer theorem which has not been revisited for almost 50 years and formulates several related centralizer problems. In the third section, we investigate a free algebra analogue of a classical theorem of Białynicki-Birula’s theorem and give a noncommutative version of this famous theorem. Additionally, we consider positive-root torus actions and obtain the linearity property analogous to the Białynicki-Birula theorem. In the last sections, we introduce Feigin’s homomorphisms and we see how they help us in proving our main and fundamental theorems on screening operators and in the construction of our lattice Wn-algebras associated with sln, which is by far the simplest known approach concerning constructing such algebras until now.

For a positive integer [Formula: see text], let [Formula: see text] be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank n. We show that the subgroup of [Formula: see text] generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of [Formula: see text] is dense in [Formula: see text] with respect to the formal power series topology.

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

This paper proves that the Nagata automorphism over a finite field can be mimicked by a tame automorphism which is a composition of four elementary automorphisms. By investigating the sign of the permutations induced by the above elementary automorphisms, one can see that if the Nagata automorphism is defined over a prime field of characteristic two, the Nagata automorphism induces an odd permutation, and otherwise, the Nagata automorphism induces an even permutation.

For positive integers n and k, with n≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 4$$\\end{document}, let Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{n}$$\\end{document} be the free group of rank n and let Gn,k=Fn/γ3(Fn′)[Fn″,kFn]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_{n,k} = F_{n}/\\gamma _{3}(F^{\\prime }_{n})[F^{\\prime \\prime }_{n},~_{k}F_{n}]$$\\end{document}. We show that for sufficiently large n, the automorphism group Aut(Gn,k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{Aut}}(G_{n,k})$$\\end{document} of Gn,k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_{n,k}$$\\end{document} is generated by the tame automorphisms and one more non-tame automorphism.

Let R be a PID. We construct and classify all coordinates of R[x, y] of the form p 2 y + Q 2(p 1 x + Q 1(y)) with p 1, p 2 ∈ qt(R) and Q 1, Q 2 ∈ qt(R)[y]. From this construction (with R = K[z]) we obtain nontame automorphisms σ of K[x, y, z] (where K is a field of characteristic 0) such that the subgroup generated by σ and the affine automorphisms contains all tame automorphisms.

The grid-based key predistribution scheme in sensor networks has some advantages over the existing approaches. However, since this scheme is based on polynomial-based approach, it cannot provide authentication service. In this paper, we first present a novel tame-based approach, in which a tame automorphism in algebra is exploited to generate a symmetric and two-one bivariate map, for key predistribution. This tame-based approach can achieve the goal of authentication. We then propose a variance of the grid-based scheme, based on this tame-based approach and utilizing deployment information. As a result, the proposed scheme can offer basic authentication service in sensor networks and have better performance, in addition to preserving the advantages of the grid-based scheme.