Existence Results for Finite Field Polynomials with Specified Properties

Abstract This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.

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Beta dynamical systems over the formal fields

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1291 ◽

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.

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Valued Field Graph and Some Related Parameters

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.30 ◽

2020 ◽

pp. 389-395

Author(s):

G. Suresh Singh ◽

P. K. Prasobha

Keyword(s):

Finite Field ◽

Independence Number ◽

Valued Field ◽

Vertex Set ◽

Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

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Existence of Gibbs point processes with stable infinite range interaction

Journal of Applied Probability ◽

10.1017/jpr.2020.39 ◽

2020 ◽

Vol 57 (3) ◽

pp. 775-791

Author(s):

David Dereudre ◽

Thibaut Vasseur

Keyword(s):

Point Processes ◽

Existence Results ◽

Pairwise Interaction ◽

Range Interaction ◽

Pairwise Interactions ◽

Gibbs Point Processes ◽

Infinite Range Interactions ◽

Finite Configuration ◽

Multi Body ◽

Infinite Range

AbstractWe provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

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On the Security of Public-Key Algorithms Based on Chebyshev Polynomials over the Finite Field $Z_N$

IEEE Transactions on Computers ◽

10.1109/tc.2010.148 ◽

2010 ◽

Vol 59 (10) ◽

pp. 1392-1401 ◽

Cited By ~ 19

Author(s):

Xiaofeng Liao ◽

Fei Chen ◽

Kwok-wo Wong

Keyword(s):

Finite Field ◽

Chebyshev Polynomials ◽

Public Key

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Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-014-0219-8 ◽

2014 ◽

Vol 17 (4) ◽

Cited By ~ 4

Author(s):

Shengli Xie

Keyword(s):

Differential Equations ◽

Differential Evolution ◽

Existence Theorem ◽

Banach Spaces ◽

Existence And Uniqueness ◽

Evolution Equations ◽

Infinite Delay ◽

Mild Solutions ◽

Existence Results ◽

Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

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