Representing Autonomous Probabilistic Automata by Minimum Characteristic Polynomials over a Finite Field

2021 ◽  
pp. 215-224
Author(s):  
Vjacheslav M. Zakharov ◽  
Sergei V. Shalagin ◽  
Bulat F. Eminov
Keyword(s):  
Finite Field ◽  
Characteristic Polynomials ◽  
Probabilistic Automata

scholarly journals On the class of square Petrie matrices induced by cyclic permutations

2004 ◽  
Vol 2004 (31) ◽  
pp. 1617-1622
Author(s):  
Bau-Sen Du
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Characteristic Polynomials ◽  
Dynamical Properties

Letn≥2be an integer and letP={1,2,…,n,n+1}. LetZpdenote the finite field{0,1,2,…,p−1}, wherep≥2is a prime. Then every mapσonPdetermines a realn×nPetrie matrixAσwhich is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization ofσ. In this paper, we show that ifσis acyclicpermutation onP, then all such matricesAσare similar to one another overZ2(but not overZpfor any primep≥3) and their characteristic polynomials overZ2are all equal to∑k=0nxk. As a consequence, we obtain that ifσis acyclicpermutation onP, then the coefficients of the characteristic polynomial ofAσare all odd integers and hence nonzero.

Linearized systems and a generalized Cayley–Hamilton theorem

2017 ◽  
Vol 16 (06) ◽  
pp. 1750120
Author(s):  
Jeffrey Lang ◽  
Daniel Newland
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Minimum Distance ◽  
Solution Space ◽  
Arbitrary Field ◽  
Characteristic Polynomials ◽  
Systems Of Equations ◽  
Square Matrix

We study linearized systems of equations in characteristic [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a square matrix and [Formula: see text]. We present algorithms for calculating their solutions and for determining the minimum distance of their solution spaces. In the case when [Formula: see text] has entries in [Formula: see text], the finite field of [Formula: see text] elements, we explore the relationships between the minimal and characteristic polynomials of [Formula: see text] and the above mentioned features of the solution space. In order to extend and generalize these findings to the case when [Formula: see text] has entries in an arbitrary field of characteristic [Formula: see text], we obtain generalizations of the characteristic polynomial of a matrix and the Cayley–Hamilton theorem to square linearized systems.

scholarly journals Computing the Characteristic Polynomials of a Class of Hyperelliptic Curves for Cryptographic Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
Lin You ◽  
Guangguo Han ◽  
Jiwen Zeng ◽  
Yongxuan Sang
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Hyperelliptic Curve ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Common Method ◽  
Characteristic Polynomials ◽  
Computational Data ◽  
Prime Factors

Hyperelliptic curves have been widely studied for cryptographic applications, and some special hyperelliptic curves are often considered to be used in practical cryptosystems. Computing Jacobian group orders is an important operation in constructing hyperelliptic curve cryptosystems, and the most common method used for the computation of Jacobian group orders is by computing the zeta functions or the characteristic polynomials of the related hyperelliptic curves. For the hyperelliptic curveCq:v2=up+au+bover the fieldFqwithqbeing a power of an odd primep, Duursma and Sakurai obtained its characteristic polynomial forq=p,a=−1,andb∈Fp. In this paper, we determine the characteristic polynomials ofCqover the finite fieldFpnforn=1, 2 anda,b∈Fpn. We also give some computational data which show that many of those curves have large prime factors in their Jacobian group orders, which are both practical and vital for the constructions of efficient and secure hyperelliptic curve cryptosystems.

scholarly journals CHARACTERISTIC POLYNOMIALS OF SIMPLE ORDINARY ABELIAN VARIETIES OVER FINITE FIELDS

2021 ◽  
pp. 1-7
Author(s):  
LENNY JONES
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Abelian Varieties ◽  
Characteristic Polynomials ◽  
Easy Method

Abstract We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties ${{\mathcal A}}$ of dimension g over a finite field ${{\mathbb F}}_q$ , when $q\ge 4$ and $2g=\rho ^{b-1}(\rho -1)$ , for some prime $\rho \ge 5$ with $b\ge 1$ . Moreover, we show that ${{\mathcal A}}$ is absolutely simple if $b=1$ and g is prime, but ${{\mathcal A}}$ is not absolutely simple for any prime $\rho \ge 5$ with $b>1$ .

scholarly journals Representing stand-alone automata by characteristic polynomials over a finite field

2020 ◽  
Vol 1658 ◽  
pp. 012077
Author(s):  
V M Zakharov ◽  
S V Shalagin ◽  
B F Eminov
Keyword(s):  
Finite Field ◽  
Characteristic Polynomials

scholarly journals Le lemme fondamental pondéré. I. Constructions géométriques

2010 ◽  
Vol 146 (6) ◽  
pp. 1416-1506 ◽  
Author(s):  
Pierre-Henri Chaudouard ◽  
Gérard Laumon
Keyword(s):  
Finite Field ◽  
Affine Space ◽  
Base Space ◽  
Characteristic Polynomials ◽  
Selberg Trace Formula ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Open Set ◽  
Simple Set ◽  
Hitchin Fibration

AbstractThis work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg trace formula.

scholarly journals The Characteristic Polynomials of Symmetric Graphs

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 582
Author(s):  
Nafaa Chbili ◽  
Shamma Al Dhaheri ◽  
Mei Tahnon ◽  
Amna Abunamous
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Finite Graph ◽  
Characteristic Polynomials ◽  
Induced Subgraph ◽  
P Elements ◽  
Fixed Set ◽  
Algebraic Invariants

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ , with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ [ F ] by one of Γ \ F . A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ .

Sign in / Sign up

Export Citation Format

Share Document