scholarly journals A lower bound on the essential dimension of a connected linear group

2009 ◽  
pp. 189-212 ◽  
Author(s):  
Philippe Gille ◽  
Zinovy Reichstein
Keyword(s):  
Lower Bound ◽  
Linear Group ◽  
Essential Dimension

scholarly journals ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC

2017 ◽  
Vol 5 ◽  
Author(s):  
KELLY MCKINNIE
Keyword(s):  
Lower Bound ◽  
Essential Dimension ◽  
Base Field ◽  
Witt Group ◽  
New Techniques ◽  
Characteristic 2 ◽  
Symbol Algebra ◽  
Bounded Below ◽  
Independent Parameters

In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$ when the base field is algebraically closed of characteristic $p$. The proof uses new techniques for working with residues in Milne–Kato $p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on $p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic $p$-symbol algebra of length $\ell$ is shown to have $p$-essential dimension equal to $\ell +1$ as a $p$-torsion Brauer class. The second is a lower bound of $\ell +1$ on the $p$-essential dimension of the functor $\operatorname{Alg}_{p^{\ell },p}$. Roughly speaking this says that you will need at least $\ell +1$ independent parameters to be able to specify any given algebra of degree $p^{\ell }$ and exponent $p$ over a field of characteristic $p$ and improves on the previously established lower bound of 3.

scholarly journals A fiber dimension theorem for essential and canonical dimension

2012 ◽  
Vol 149 (1) ◽  
pp. 148-174 ◽  
Author(s):  
Roland Lötscher
Keyword(s):  
Algebraic Geometry ◽  
Lower Bound ◽  
Finite Type ◽  
Algebraic Groups ◽  
General Version ◽  
Essential Dimension ◽  
Algebraic Stacks ◽  
Algebraic Tori ◽  
Canonical Dimension

AbstractThe well-known fiber dimension theorem in algebraic geometry says that for every morphism f:X→Y of integral schemes of finite type the dimension of every fiber of f is at least dim X−dim Y. This has recently been generalized by Brosnan, Reichstein and Vistoli to certain morphisms of algebraic stacks f:𝒳→𝒴, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover, we will prove a variant of this theorem, where essential dimension and canonical dimension are linked. These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by MacDonald, Meyer, Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

scholarly journals A lower bound on the essential dimension of PGL4 in characteristic 2

2017 ◽  
Vol 16 (04) ◽  
pp. 1750063
Author(s):  
Sanghoon Baek
Keyword(s):  
Lower Bound ◽  
Cohomology Group ◽  
Positive Characteristic ◽  
Essential Dimension ◽  
Trace Form ◽  
Simple Algebras ◽  
Characteristic 2 ◽  
Field Of Positive Characteristic

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato’s cohomology group, defined by cokernel of a general Artin–Schreier operator. Combining this with Tignol’s result on the second trace form of simple algebras of degree [Formula: see text], we show that [Formula: see text] over a field of characteristic [Formula: see text].

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

2019 ◽  
Vol 485 (2) ◽  
pp. 142-144
Author(s):  
A. A. Zevin
Keyword(s):  
Lower Bound ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Lipschitz Constant ◽  
Minimal Period ◽  
Arbitrary Vector ◽  
Vector Norm ◽  
Sharp Lower Bound

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

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