Fractal mathematical over extended finite fields Fp[x]/(f(x))

In this paper, we have defined an algorithm for the construction of iterative operations, based on dimensional projections and correspondence between the properties of extended fields, with respect to modular reduction. For a field with product operations R(x) ⊗ D(x), over finite fields, GF[(pm)n−k]. With Gp[x]/(g(f(x)), whence the coefficient of the g(x) is replaced after a modular reduction operation, with characteristic p. Thus, the reduced coefficients of the generating polynomial of G contain embedded the modular reduction and thus simplify operations that contain basic finite fields. The algorithm describes the process of construction of the GF multiplier, it can start at any stage of LFSR; it is shift the sequence of operation, from this point on, thanks to the concurrent adaptation, to optimize the energy consumption of the GF iterative multiplier circuit, we can claim that this method is more efficient. From this, it was realized the mathematical formalization of the characteristics of the iterative operations on the extended finite fields has been developed, we are applying a algorithm several times over the coefficients in the smaller field and then in the extended field, concurrent form.

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Sylow $p$-subgroups of the general linear group over finite fields of characteristic $p$

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1955-0069175-3 ◽

1955 ◽

Vol 6 (3) ◽

pp. 454-454

Author(s):

A. J. Weir

Keyword(s):

Linear Group ◽

Finite Fields ◽

General Linear Group ◽

General Linear ◽

Characteristic P

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Computing p th roots in extended finite fields of prime characteristic p ≥ 2

Electronics Letters ◽

10.1049/el.2015.4141 ◽

2016 ◽

Vol 52 (9) ◽

pp. 718-719

Author(s):

M. Repka

Keyword(s):

Finite Fields ◽

Prime Characteristic ◽

Characteristic P

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Fermat Jacobians of Prime Degree over Finite Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-1999-009-7 ◽

1999 ◽

Vol 42 (1) ◽

pp. 78-86 ◽

Cited By ~ 3

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.

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The Number of Rational Points of a Family of Algebraic Varieties over Finite Fields

Algebra Colloquium ◽

10.1142/s1005386717000475 ◽

2017 ◽

Vol 24 (04) ◽

pp. 705-720 ◽

Cited By ~ 4

Author(s):

Shuangnian Hu ◽

Junyong Zhao

Keyword(s):

Number Theory ◽

Normal Form ◽

Finite Fields ◽

Rational Points ◽

System Of Equations ◽

Algebraic Varieties ◽

Natural Conditions ◽

Characteristic P ◽

Exponent Matrix ◽

Positive Integers

Let 𝔽q stand for the finite field of odd characteristic p with q elements (q = pn, n ∈ ℕ) and [Formula: see text] denote the set of all the nonzero elements of 𝔽q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over [Formula: see text] where the integers t > 0, r0 = 0 < r1 < r2 < ⋯ < rt, 1 ≤ n1 < n2 <, ⋯ < nt and 0 ≤ j ≤ t − 1, bk ∊ 𝔽q, ak,i ∊ [Formula: see text] (k = 1, …, m, i = 1, …, rt), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 𝔽q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015) 135–153].

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Certain 3-decompositions of complete graphs, with an application to finite fields

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500014293 ◽

1985 ◽

Vol 99 (3-4) ◽

pp. 277-281 ◽

Cited By ~ 3

Author(s):

Peter Rowlinson

Keyword(s):

Finite Field ◽

Finite Fields ◽

Complete Graph ◽

Disjoint Union ◽

Necessary Condition ◽

Complete Graphs ◽

Characteristic P ◽

Strongly Regular

SynopsisA necessary condition is obtained for a complete graph to have a decomposition as the line-disjoint union of three isomorphic strongly regular subgraphs. The condition is used to determine the number of non-trivial solutions of the equation x3+y3 = z3 in a finite field of characteristic p ≡ 2 mod 3.

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On Katz’s (A,B)-exponential sums

The Quarterly Journal of Mathematics ◽

10.1093/qmath/haaa045 ◽

2020 ◽

Author(s):

Lei FU ◽

Daqing WAN

Keyword(s):

Finite Fields ◽

Arithmetic Progression ◽

Betti Number ◽

Exponential Sums ◽

Decomposition Theorem ◽

Characteristic P ◽

Universal Family ◽

Degenerate Cases

Abstract We deduce Katz’s theorems for (A, B)-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef–Loeser, removing the hypothesis that A + B is relatively prime to the characteristic p. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson–Sperber’s bounds for degrees of L-functions. Applying the facial decomposition theorem, we prove that the universal family of (A, B)-polynomials is generically ordinary for its L-function when p is in certain arithmetic progression.

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