scholarly journals On the surjectivity of the power maps of algebraic groups in characteristic zero

2002 ◽  
Vol 9 (6) ◽  
pp. 741-756 ◽  
Author(s):  
Pralay Chatterjee
Keyword(s):  
Characteristic Zero ◽  
Algebraic Groups ◽  
Power Maps

scholarly journals DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

2015 ◽  
Vol 16 (1) ◽  
pp. 59-119 ◽  
Author(s):  
Lucia Di Vizio ◽  
Charlotte Hardouin ◽  
Michael Wibmer
Keyword(s):  
Differential Equations ◽  
Characteristic Zero ◽  
Galois Theory ◽  
Structure Theorem ◽  
Special Functions ◽  
Algebraic Groups ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Linear Differential

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

scholarly journals Méthodes fonctionnelles pour la transcendance en caractéristique finie

1994 ◽  
Vol 50 (2) ◽  
pp. 273-286 ◽  
Author(s):  
Laurent Denis
Keyword(s):  
Exponential Function ◽  
Characteristic Zero ◽  
Algebraic Closure ◽  
Algebraic Groups ◽  
Zero Element ◽  
Finite Characteristic ◽  
First Derivative ◽  
Linearly Independent ◽  
Divisibility Properties

There are essentially two ways to obtain transcendence results in finite characteristic. The first, historically, is to use Ore's lemma and to prove that a series whose coefficients satisfy well-behaved divisibility properties cannot be a zero of an additive polynomial. This method is of the same kind as the method of p–automata. The second one is to try to imitate the usual methods in characteristic zero and to do transcendence theory with t–modules analogously to what we can do with algebraic groups. We want to show here that transcendence results over Fq(T) can also be obtained with the help of the variable T. If ec(z) is the Carlitz exponential function and e = ec(1), we obtain, in particular, that 1, e, …, e(p–2) (the P–2 first derivative of e with respect to T) are linearly independent over the algebraic closure of Fq(T). A corollary is that for every non-zero element α in Fq((1/T)), αpe and αec(e1/p) are transcendental over Fq(T). By changing the variable and using older results we also obtain the transcendence of ec(ω) for all ω ∈ Fq((1/T)) such that ω(T) and ω(Ti) are not zero and linearly dependent over Fq (Ti) (q > 2i + 1). Such u appear to be transcendental by the method of Mahler if i is not a power of p.

scholarly journals Zero-Separating Invariants for Linear Algebraic Groups

2015 ◽  
Vol 59 (4) ◽  
pp. 911-924 ◽  
Author(s):  
Jonathan Elmer ◽  
Martin Kohls
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Elementary Proof ◽  
Algebraic Groups ◽  
Linear Algebraic Group ◽  
Null Cone ◽  
Closed Field ◽  
Unipotent Group ◽  
Linear Algebraic Groups ◽  
Separating Invariants

AbstractAbstract Let G be a linear algebraic group over an algebraically closed field 𝕜 acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and , respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL2(𝕜) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G0 is unipotent. Our results also lead to a more elementary proof that βsep(G) is finite if and only if G is finite.

scholarly journals Surjectivity of power maps of real algebraic groups

2011 ◽  
Vol 226 (6) ◽  
pp. 4639-4666 ◽  
Author(s):  
Pralay Chatterjee
Keyword(s):  
Algebraic Groups ◽  
Power Maps

scholarly journals On the Popov–Pommerening conjecture for linear algebraic groups

2017 ◽  
Vol 154 (1) ◽  
pp. 36-79
Author(s):  
Gergely Bérczi
Keyword(s):  
Characteristic Zero ◽  
Maximal Torus ◽  
Algebraic Groups ◽  
Reductive Group ◽  
Affirmative Answer ◽  
Classical Groups ◽  
Finitely Generated ◽  
Linear Algebraic Groups ◽  
Invariant Algebra

Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H\subseteq G$ an observable subgroup normalised by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov and Pommerening conjectured in the late 1970s that the invariant algebra $k[X]^{H}$ is finitely generated. We prove the conjecture for: (1) subgroups of $\operatorname{SL}_{n}(k)$ closed under left (or right) Borel action and for: (2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of $\operatorname{SL}_{n}(k)$.

scholarly journals p-power conjugacy classes in U(n,q) and T(n,q)

2020 ◽  
pp. 2150121
Author(s):  
Silvio Dolfi ◽  
Anupam Singh ◽  
Manoj K. Yadav
Keyword(s):  
Algebraic Groups ◽  
Recursive Formula ◽  
Conjugacy Classes ◽  
Image Size ◽  
Large Size ◽  
Power Maps ◽  
Unitriangular Group ◽  
Triangular Group

Let [Formula: see text] be a [Formula: see text]-power where [Formula: see text] is a fixed prime. In this paper, we look at the [Formula: see text]-power maps on unitriangular group [Formula: see text] and triangular group [Formula: see text]. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.

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