scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

Fermat Jacobians of Prime Degree over Finite Fields

1999 ◽  
Vol 42 (1) ◽  
pp. 78-86 ◽  
Author(s):  
Josep González
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Cyclotomic Field ◽  
Roots Of Unity ◽  
Characteristic P ◽  
Prime Degree ◽  
Numerical Criterion

AbstractWe study the splitting of Fermat Jacobians of prime degree l over an algebraic closure of a finite field of characteristic p not equal to l. We prove that their decomposition is determined by the residue degree of p in the cyclotomic field of the l-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

scholarly journals On plane curves given by separated polynomials and their automorphisms

2020 ◽  
Vol 20 (1) ◽  
pp. 61-70
Author(s):  
Matteo Bonini ◽  
Maria Montanucci ◽  
Giovanni Zini
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Algebraic Closure ◽  
Sufficient Conditions ◽  
Plane Curve ◽  
Plane Curves ◽  
Prime Field ◽  
Full Automorphism Group ◽  
Ag Codes ◽  
Finite Prime Field

AbstractLet 𝓒 be a plane curve defined over the algebraic closure K of a finite prime field 𝔽p by a separated polynomial, that is 𝓒 : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of 𝓒 when m ≢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of 𝓒 to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve 𝓒 : X(qr – 1)/(q–1) = Yqr–1 + Yqr–2 + … + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms.

scholarly journals GEOMETRIC REPRESENTATION IN THE THEORY OF PSEUDO-FINITE FIELDS

2017 ◽  
Vol 82 (3) ◽  
pp. 1132-1139 ◽  
Author(s):  
ÖZLEM BEYARSLAN ◽  
ZOÉ CHATZIDAKIS
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Geometric Representation

AbstractWe study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

scholarly journals On the number of permutation polynomials over a finite field

2016 ◽  
Vol 12 (06) ◽  
pp. 1519-1528
Author(s):  
Kwang Yon Kim ◽  
Ryul Kim ◽  
Jin Song Kim
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Roots Of Unity ◽  
Permutation Polynomials

In order to extend the results of [Formula: see text] in [P. Das, The number of permutation polynomials of a given degree over a finite field, Finite Fields Appl. 8(4) (2002) 478–490], where [Formula: see text] is a prime, to arbitrary finite fields [Formula: see text], we find a formula for the number of permutation polynomials of degree [Formula: see text] over a finite field [Formula: see text], which has [Formula: see text] elements, in terms of the permanent of a matrix. We write down an expression for the number of permutation polynomials of degree [Formula: see text] over a finite field [Formula: see text], using the permanent of a matrix whose entries are [Formula: see text]th roots of unity and using this we obtain a nontrivial bound for the number. Finally, we provide a formula for the number of permutation polynomials of degree [Formula: see text] less than [Formula: see text].

scholarly journals GENERALIZED GELFAND–GRAEV REPRESENTATIONS IN SMALL CHARACTERISTICS

2016 ◽  
Vol 224 (1) ◽  
pp. 93-167 ◽  
Author(s):  
JAY TAYLOR
Keyword(s):  
Finite Field ◽  
Wave Front ◽  
Irreducible Character ◽  
Prime Order ◽  
Algebraic Closure ◽  
Characteristic Functions ◽  
Rational Structure ◽  
Reductive Algebraic Group ◽  
Wave Front Set ◽  
Frobenius Endomorphism

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_{p}}$ of the finite field of prime order $p$ and let $F:\mathbf{G}\rightarrow \mathbf{G}$ be a Frobenius endomorphism with $G=\mathbf{G}^{F}$ the corresponding $\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139–179) is only valid under the assumption that $p$ is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, respectively, character sheaf of $\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever $p$ is good for $\mathbf{G}$.

scholarly journals Graph polynomials and symmetries

2019 ◽  
Vol 18 (09) ◽  
pp. 1950172 ◽  
Author(s):  
Nafaa Chbili
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Prime Order ◽  
Tutte Polynomial ◽  
Quotient Graph ◽  
Graph Polynomials ◽  
Tutte Polynomials

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

MULTIPLICATIVE ORDERS IN ORBITS OF POLYNOMIALS OVER FINITE FIELDS

2017 ◽  
Vol 60 (2) ◽  
pp. 487-493 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Fields ◽  
Algebraic Closure ◽  
Rational Numbers ◽  
Duke Math ◽  
Roots Of Unity ◽  
Multiplicative Order ◽  
Polynomials Over Finite Fields ◽  
Polynomial Orbits

AbstractWe show, under some natural restrictions, that orbits of polynomials cannot contain too many elements of small multiplicative order modulo a large prime p. We also show that for all but finitely many initial points either the multiplicative order of this point or the length of the orbit it generates (both modulo a large prime p) is large. The approach is based on the results of Dvornicich and Zannier (Duke Math. J.139 (2007), 527–554) and Ostafe (2017) on roots of unity in polynomial orbits over the algebraic closure of the field of rational numbers.

Maximal T-spaces of free associative algebras over a finite field

2018 ◽  
Vol 17 (04) ◽  
pp. 1850064
Author(s):  
C. Bekh-Ochir ◽  
S. A. Rankin
Keyword(s):  
Finite Field ◽  
Positive Characteristic ◽  
Associative Algebras ◽  
Prime Field ◽  
Characteristic 2 ◽  
Infinite Set

In earlier work, it was established that for any finite field [Formula: see text] and any nonempty set [Formula: see text], the free associative (nonunitary) [Formula: see text]-algebra on [Formula: see text], denoted by [Formula: see text], had infinitely many maximal [Formula: see text]-spaces, but exactly two maximal [Formula: see text]-ideals (each of which was shown to be a maximal [Formula: see text]-space). This raises the interesting question as to whether or not the maximal [Formula: see text]-spaces can be classified. However, aside from the two maximal [Formula: see text]-ideals, no examples of maximal [Formula: see text]-spaces of [Formula: see text] have been identified to this point. This paper presents, for each finite field [Formula: see text], an infinite set of proper [Formula: see text]-spaces [Formula: see text] of [Formula: see text], none of which is a [Formula: see text]-ideal. It is proven that for any distinct integers [Formula: see text], [Formula: see text]. Furthermore, it is proven that for the prime field [Formula: see text], [Formula: see text] any prime, [Formula: see text] is a maximal [Formula: see text]-space of [Formula: see text]. We conjecture that for any finite field [Formula: see text] of positive characteristic different from 2 and each integer [Formula: see text], [Formula: see text] is a maximal [Formula: see text]-space of [Formula: see text]. In characteristic 2, the situation is slightly different and we provide different candidates for maximal [Formula: see text]-spaces.

Strong map-symmetry of SL(3,K) and PSL(3,K) for every finite field K

2020 ◽  
pp. 2150048
Author(s):  
Antonio Breda d’Azevedo ◽  
Domenico A. Catalano
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Automorphism Groups ◽  
Group Automorphism ◽  
Regular Maps ◽  
Strong Map

In this paper, we show that for any finite field [Formula: see text], any pair of map-generators (that is when one of the generators is an involution) of [Formula: see text] and [Formula: see text] has a group automorphism that inverts both generators. In the theory of maps, this corresponds to say that any regular oriented map with automorphism group [Formula: see text] or [Formula: see text] is reflexible, or equivalently, there are no chiral regular maps with automorphism group [Formula: see text] or [Formula: see text]. As remarked by Leemans and Liebeck, also [Formula: see text] and [Formula: see text] are not automorphism groups of chiral regular maps. These two results complete the work of the above authors on simples groups supporting chiral regular maps.

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