Some co-tame automorphisms of affine spaces

2021 ◽  
pp. 1-12
Author(s):  
Dayan Liu ◽  
Fumei Liu ◽  
Xiaosong Sun
Keyword(s):  
Affine Space ◽  
Small Dimension ◽  
Transcendence Degree ◽  
Affine Spaces

The investigation of co-tame automorphisms of the affine space [Formula: see text] is helpful to understand the structure of its automorphisms group. In this paper, we show the co-tameness of several classes of automorphisms, including some 3-parabolic automorphisms, power-linear automorphisms, homogeneous automorphisms in small dimension or small transcendence degree. We also classify all additive-nilpotent automorphisms in dimension four and show that they are co-tame.

scholarly journals Dynamical degrees of affine-triangular automorphisms of affine spaces

2021 ◽  
pp. 1-42
Author(s):  
JÉRÉMY BLANC ◽  
IMMANUEL VAN SANTEN
Keyword(s):  
Affine Space ◽  
Affine Spaces ◽  
Dynamical Degree ◽  
Perron Number ◽  
Dynamical Degrees ◽  
Affine Automorphism

Abstract We study the possible dynamical degrees of automorphisms of the affine space $\mathbb {A}^n$ . In dimension $n=3$ , we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb {A}^n$ for some n, and we give the best possible n for quadratic integers, which is either $3$ or $4$ .

scholarly journals Quotient complete intersections of affine spaces by finite linear groups

1985 ◽  
Vol 98 ◽  
pp. 1-36 ◽  
Author(s):  
Haruhisa Nakajima
Keyword(s):  
Normal Subgroup ◽  
Affine Space ◽  
Finite Subgroup ◽  
Complete Intersections ◽  
Linear Groups ◽  
Coordinate Ring ◽  
Affine Spaces ◽  
Quotient Variety ◽  
Finite Linear ◽  
Complex Number Field

Let G be a finite subgroup of GLn(C) acting naturally on an affine space Cn of dimension n over the complex number field C and denote by Cn/G the quotient variety of Cn under this action of G. The purpose of this paper is to determine G completely such that Cn/G is a complete intersection (abbrev. CI.) i.e. its coordinate ring is a C.I. when n > 10. Our main result is (5.1). Since the subgroup N generated by all pseudo-reflections in G is a normal subgroup of G and Cn/G is obtained as the quotient variety of without loss of generality, we may assume that G is a subgroup of SLn(C) (cf. [6, 16, 24, 25]).

On Complete Intersections Over an Algebraically Non-Closed Field

1986 ◽  
Vol 29 (2) ◽  
pp. 140-145 ◽  
Author(s):  
Maria Grazia Marinari ◽  
Mario Raimondo
Keyword(s):  
Complete Intersection ◽  
Affine Space ◽  
Complete Intersections ◽  
Closed Field ◽  
Affine Variety ◽  
Affine Spaces ◽  
Real Curve

AbstractWe give a criterion in order that an affine variety defined over any field has a complete intersection (ci.) embedding into some affine space. Moreover we give an example of a smooth real curve C all of whose embeddings into affine spaces are c.i.; nevertheless it has an embedding into ℝ3 which cannot be realized as a c.i. by polynomials.

scholarly journals Commuting matrices and the Hilbert scheme of points on affine spaces

2018 ◽  
Vol 18 (4) ◽  
pp. 467-482
Author(s):  
Abdelmoubine A. Henni ◽  
Marcos Jardim
Keyword(s):  
Hilbert Scheme ◽  
Affine Space ◽  
Affine Spaces ◽  
Commuting Matrices ◽  
Hilbert Scheme Of Points

Abstract We give linear algebraic and monadic descriptions of the Hilbert scheme of points on the affine space of dimension n which naturally extends Nakajima’s representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of c points on ℂ3 is irreducible for c ≤ 10.

A Normal Form and the Term-linearity of Affine Spaces over GF(3)

2006 ◽  
Vol 13 (04) ◽  
pp. 675-684
Author(s):  
Jung R. Cho ◽  
Woo Hyun Kim
Keyword(s):  
Normal Form ◽  
Word Problem ◽  
Affine Space ◽  
Linear Term ◽  
Binary Trees ◽  
Vector Spaces ◽  
Affine Spaces ◽  
Term Linear

An algebra is called term-linear if every term is equal to a linear term or a constant and every n-ary linear term is essential. The term-linearity is a generalization of the linearity of polynomials over ordinary linear algebras or vector spaces. In this paper, as a sequel to a previous paper, we present an explicit normal form for terms over an affine space over GF(3) by binary trees, which shows that the word problem for affine spaces over GF(3) is solvable. Using this normal form, we count the number of all essentially n-ary linear terms and apply the Csákány formula to show that an affine space over GF(3) is term-linear.

scholarly journals Commutative algebraic monoid structures on affine spaces

2019 ◽  
Vol 22 (08) ◽  
pp. 1950064
Author(s):  
Ivan Arzhantsev ◽  
Sergey Bragin ◽  
Yulia Zaitseva
Keyword(s):  
Characteristic Zero ◽  
Affine Space ◽  
Arbitrary Dimension ◽  
Toric Varieties ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Affine Spaces ◽  
Algebraic Monoid ◽  
Algebraic Monoids

We study commutative associative polynomial operations [Formula: see text] with unit on the affine space [Formula: see text] over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

scholarly journals Characterizations of Finite Projective and Affine Spaces

1969 ◽  
Vol 21 ◽  
pp. 64-75 ◽  
Author(s):  
William M. Kantor
Keyword(s):  
Affine Space ◽  
Incidence Structure ◽  
Principal Result ◽  
List Type ◽  
Affine Spaces ◽  
Finite Projective Spaces ◽  
Finite Incidence ◽  
Finite Incidence Structure ◽  
Positive Integers ◽  
Projective And Affine Spaces

A well-known result of Dembowski and Wagner (4) characterizes the designs of points and hyperplanes of finite projective spaces among all symmetric designs. By passing to a dual situation and approaching this idea from a different direction, we shall obtain common characterizations of finite projective and affine spaces. Our principal result is the following.Theorem 1.A finite incidence structure is isomorphic to the design of points and hyperplanes of a finite projective or affine space of dimension greater than or equal to4if and only if there are positive integers v, k, and y, with μ> 1and(μ– l)(v — k) ≠ (k—μ)2such that the following assumptions hold.(I)Every block is on k points, and every two intersecting blocks are on μ common points.(II)Given a point and two distinct blocks, there is a block containing both the point and the intersection of the blocks.(III)Given two distinct points p and q, there is a block on p but not on q.(IV)There are v points, and v– 2 ≧k>μ.

scholarly journals Points of Small Height on Varieties Defined over a Function Field

2009 ◽  
Vol 52 (2) ◽  
pp. 237-244
Author(s):  
Dragos Ghioca
Keyword(s):  
Finite Field ◽  
Function Field ◽  
Affine Space ◽  
Algebraic Closure ◽  
Transcendence Degree ◽  
Small Height

AbstractWe obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree 1 over a finite field.

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