Fast arithmetic for the algebraic closure of finite fields

2014 ◽  
Author(s):  
Luca De Feo ◽  
Javad Doliskani ◽  
Éric Schost
Keyword(s):  
Finite Fields ◽  
Algebraic Closure ◽  
Fast Arithmetic

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.

scholarly journals IITAKA CONJECTURE FOR 3-FOLDS OVER FINITE FIELDS

2016 ◽  
Vol 229 ◽  
pp. 21-51 ◽  
Author(s):  
CAUCHER BIRKAR ◽  
YIFEI CHEN ◽  
LEI ZHANG
Keyword(s):  
Finite Fields ◽  
Algebraic Closure ◽  
Birational Geometry ◽  
Minimal Models ◽  
The Way

We prove Iitaka $C_{n,m}$ conjecture for $3$-folds over the algebraic closure of finite fields. Along the way we prove some results on the birational geometry of log surfaces over nonclosed fields and apply these to existence of relative good minimal models of $3$-folds.

scholarly journals The factorable core of polynomials over finite fields

1990 ◽  
Vol 49 (2) ◽  
pp. 309-318 ◽  
Author(s):  
S. D. Cohen
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Permutation Polynomials ◽  
Linearized Polynomial ◽  
Polynomials Over Finite Fields ◽  
Value Sets

AbstractFor a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned.

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.

A link between minimal value set polynomials and tamely ramified towers of function fields over finite fields

2021 ◽  
pp. 2250189
Author(s):  
R. Toledano
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Finite Subset ◽  
Function Fields ◽  
Algebraic Closure ◽  
Value Set ◽  
Towers Of Function Fields ◽  
Value Sets

In this paper, we introduce the notions of [Formula: see text]-polynomial and [Formula: see text]-minimal value set polynomial where [Formula: see text] is a polynomial over a finite field [Formula: see text] and [Formula: see text] is a finite subset of an algebraic closure of [Formula: see text]. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are [Formula: see text]-minimal value set polynomials for [Formula: see text], whose [Formula: see text]-value sets can be explicitly computed in terms of the monomial [Formula: see text].

scholarly journals On the units generated by Weierstrass forms

2014 ◽  
Vol 17 (A) ◽  
pp. 303-313
Author(s):  
Ömer Küçüksakallı
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Curves Over Finite Fields ◽  
Elliptic Units ◽  
Fast Arithmetic

AbstractThere is an algorithm of Schoof for finding divisors of class numbers of real cyclotomic fields of prime conductor. In this paper we introduce an improvement of the elliptic analogue of this algorithm by using a subgroup of elliptic units given by Weierstrass forms. These elliptic units which can be expressed in terms of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x$-coordinates of points on elliptic curves enable us to use the fast arithmetic of elliptic curves over finite fields.

Spectra of Boolean Graphs Over Finite Fields of Characteristic Two

2019 ◽  
Vol 63 (1) ◽  
pp. 58-65
Author(s):  
D. Scott Dillery ◽  
John D. LaGrange
Keyword(s):  
Number Theory ◽  
Finite Fields ◽  
Adjacency Matrix ◽  
Zero Divisor ◽  
Algebraic Closure ◽  
Simple Graph ◽  
Boolean Ring ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Characteristic Two

AbstractWith entries of the adjacency matrix of a simple graph being regarded as elements of $\mathbb{F}_{2}$, it is proved that a finite commutative ring $R$ with $1\neq 0$ is a Boolean ring if and only if either $R\in \{\mathbb{F}_{2},\mathbb{F}_{2}\times \mathbb{F}_{2}\}$ or the eigenvalues (in the algebraic closure of $\mathbb{F}_{2}$) corresponding to the zero-divisor graph of $R$ are precisely the elements of $\mathbb{F}_{4}\setminus \{0\}$ . This is achieved by observing a way in which algebraic behavior in a Boolean ring is encoded within Pascal’s triangle so that computations can be carried out by appealing to classical results from number theory.

scholarly journals Global Norm-Residue Map over Quasi-Finite Field

1966 ◽  
Vol 27 (1) ◽  
pp. 323-329 ◽  
Author(s):  
D. S. Rim ◽  
G. Whaples
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Local Field ◽  
Algebraic Closure ◽  
Residue Field ◽  
Additive Group ◽  
Local Fields ◽  
Ordinary Sense ◽  
Reciprocity Law ◽  
Norm Residue

A field k is called quasi-finite if it is perfect and if Gk≈Ż where Gk is the Galois group of the algebraic closure kc over k and Ż is the completion of the additive group of the rational integers. The classical reciprocity law on the local field with finite residue field is well-known to hold on local fields with quasi-finite residue field ([4] [5]). Thus it is natural to ask if the global reciprocity law should hold in the ordinary sense (see § 1 below) on the function-fields of one variable over quasi-finite field. We consider here two basic prototypes of non-finite quasi-finite fields:

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