scholarly journals On Deciding Finiteness for Matrix Groups over Fields of Positive Characteristic

2001 ◽  
Vol 4 ◽  
pp. 64-72 ◽  
Author(s):  
A. Detinko
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Deterministic Algorithm ◽  
Transcendence Degree ◽  
Matrix Group ◽  
Finitely Generated ◽  
Matrix Groups ◽  
Field Of Positive Characteristic

AbstractThe author considers the development of algorithms for deciding whether a finitely generated matrix group over a field of positive characteristic is finite. A deterministic algorithm for deciding the finiteness is presented for the case of a field of transcendence degree one over a finite field.

scholarly journals Computing a Chief Series and the Soluble Radical of a Matrix Group Over a Finite Field

2008 ◽  
Vol 11 ◽  
pp. 223-251 ◽  
Author(s):  
Derek F. Holt ◽  
Mark J. Stather
Keyword(s):  
Finite Field ◽  
Matrix Group ◽  
Matrix Groups ◽  
Generating Set

AbstractWe describe an algorithm for computing a chief series, the soluble radical, and two other characteristic subgroups of a matrix group over a finite field, which is intended for matrix groups that are too large for the use of base and strong generating set methods. The algorithm has been implemented in Magma by the second author.

scholarly journals Semiample perturbations for log canonical varieties over an F-finite field containing an infinite perfect field

2017 ◽  
Vol 28 (05) ◽  
pp. 1750030 ◽  
Author(s):  
Hiromu Tanaka
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Ample Divisor ◽  
Perfect Field ◽  
Log Canonical ◽  
Canonical Pair ◽  
Field Of Positive Characteristic

Let [Formula: see text] be an [Formula: see text]-finite field containing an infinite perfect field of positive characteristic. Let [Formula: see text] be a projective log canonical pair over [Formula: see text]. In this note, we show that, for a semi-ample divisor [Formula: see text] on [Formula: see text], there exists an effective [Formula: see text]-divisor [Formula: see text] such that [Formula: see text] is log canonical if there exists a log resolution of [Formula: see text].

scholarly journals BERNSTEIN–SATO POLYNOMIALS AND TEST MODULES IN POSITIVE CHARACTERISTIC

2016 ◽  
Vol 222 (1) ◽  
pp. 74-99 ◽  
Author(s):  
MANUEL BLICKLE ◽  
AXEL STÄBLER
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Regular Variety ◽  
Structure Sheaf ◽  
Analytic Case ◽  
Nonzero Divisor ◽  
Test Ideal ◽  
Field Of Positive Characteristic

In analogy with the complex analytic case, Mustaţă constructed (a family of) Bernstein–Sato polynomials for the structure sheaf${\mathcal{O}}_{X}$and a hypersurface$(f=0)$in$X$, where$X$is a regular variety over an$F$-finite field of positive characteristic (see Mustaţă,Bernstein–Sato polynomials in positive characteristic, J. Algebra321(1) (2009), 128–151). He shows that the suitably interpreted zeros of his Bernstein–Sato polynomials correspond to the$F$-jumping numbers of the test ideal filtration${\it\tau}(X,f^{t})$. In the present paper we generalize Mustaţă’s construction replacing${\mathcal{O}}_{X}$by an arbitrary$F$-regular Cartier module$M$on$X$and show an analogous correspondence of the zeros of our Bernstein–Sato polynomials with the jumping numbers of the associated filtration of test modules${\it\tau}(M,f^{t})$provided that$f$is a nonzero divisor on$M$.

THE RAMIFICATION GROUPS AND DIFFERENT OF A COMPOSITUM OF ARTIN–SCHREIER EXTENSIONS

2010 ◽  
Vol 06 (07) ◽  
pp. 1541-1564 ◽  
Author(s):  
QINGQUAN WU ◽  
RENATE SCHEIDLER
Keyword(s):  
Class Number ◽  
Function Field ◽  
Positive Characteristic ◽  
Field Extension ◽  
Constant Field ◽  
Divisor Class ◽  
Characteristic P ◽  
Similar Nature ◽  
Field Of Positive Characteristic ◽  
The Ideal

Let K be a function field over a perfect constant field of positive characteristic p, and L the compositum of n (degree p) Artin–Schreier extensions of K. Then much of the behavior of the degree pn extension L/K is determined by the behavior of the degree p intermediate extensions M/K. For example, we prove that a place of K totally ramifies/is inert/splits completely in L if and only if it totally ramifies/is inert/splits completely in every M. Examples are provided to show that all possible decompositions are in fact possible; in particular, a place can be inert in a non-cyclic Galois function field extension, which is impossible in the case of a number field. Moreover, we give an explicit closed form description of all the different exponents in L/K in terms of those in all the M/K. Results of a similar nature are given for the genus, the regulator, the ideal class number and the divisor class number. In addition, for the case n = 2, we provide an explicit description of the ramification group filtration of L/K.

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

2021 ◽  
pp. 2250191
Author(s):  
Merrick Cai ◽  
Daniil Kalinov
Keyword(s):  
Hilbert Series ◽  
Positive Characteristic ◽  
Closed Field ◽  
Simple Criterion ◽  
Characteristic P ◽  
Polynomial Representation ◽  
Cherednik Algebra ◽  
Rational Cherednik Algebra ◽  
Polynomial Formula ◽  
Field Of Positive Characteristic

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Galois Extension ◽  
Characteristic Zero ◽  
Rational Point ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Infinite Rank ◽  
Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

scholarly journals Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

Matrix Group Actions Transforming Patterns

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Coordinate System ◽  
Spatial Patterns ◽  
Group Actions ◽  
Temporal Patterns ◽  
Matrix Group ◽  
Matrix Groups ◽  
Pattern Theory ◽  
Complex Patterns

Thus far Pattern theory has been combinatore constructing complex patterns by connecting simpler ones via graphs. Patterns typically occurring in nature may be extremely complex and exhibit invariances. For example, spatial patterns may live in a space where the choice of coordinate system is irrelevant; temporal patterns may exist independently of where time is counted from, and so on. For this matrix groups as transformations are introduced, these transformations often forming groups which act on the generators.

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