scholarly journals On the lower bound for diameter of commuting graph of prime-square sized matrices

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5993-6000
Author(s):  
David Dolzan ◽  
Damjana Kokol-Bukovsek ◽  
Bojan Kuzma
Keyword(s):  
Lower Bound ◽  
Finite Fields ◽  
Galois Extension ◽  
Rational Numbers ◽  
Large Field ◽  
Commuting Graph ◽  
Cyclic Galois Extension

It is known that the diameter of commuting graph of n-by-n matrices is bounded above by six if the graph is connected. In the commuting graph of p2-by-p2 matrices over a sufficiently large field which admits a cyclic Galois extension of degree p2 we construct two matrices at distance at least five. This shows that five is the lower bound for its diameter. Our results are applicable for all sufficiently large finite fields as well as for the field of rational numbers.

scholarly journals Distribution of points on Abelian covers over finite fields

2018 ◽  
Vol 14 (05) ◽  
pp. 1375-1401 ◽  
Author(s):  
Patrick Meisner
Keyword(s):  
Finite Fields ◽  
Galois Extension ◽  
Random Variables ◽  
Families Of Curves ◽  
Abelian Covers ◽  
Distribution Of Points ◽  
Genus Formula

We determine in this paper the distribution of the number of points on the covers of [Formula: see text] such that [Formula: see text] is a Galois extension and [Formula: see text] is abelian when [Formula: see text] is fixed and the genus, [Formula: see text], tends to infinity. This generalizes the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over [Formula: see text]. In all cases, the distribution is given by a sum of [Formula: see text] random variables.

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 A recognition algorithm for non-generic classical groups over finite fields

1999 ◽  
Vol 67 (2) ◽  
pp. 223-253 ◽  
Author(s):  
Azice C. Niemeyer ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Special Linear Group ◽  
Theoretical Background ◽  
Recognition Algorithm ◽  
Large Field ◽  
Matrix Group ◽  
Classical Groups ◽  
Original Algorithm ◽  
The Matrix

AbstractIn a previous paper the authors described an algorithm to determine whether a group of matrices over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm was designed to work for all sufficiently large field sizes and dimensions of the matrix group. However, it did not apply to certain small cases. Here we present an algorithm to handle the remaining cases. The theoretical background of the algorithm presented in this paper is a substantial extension of that needed for the original algorithm.

scholarly journals Bounded realization of l-groups over global fields

1998 ◽  
Vol 150 ◽  
pp. 13-62 ◽  
Author(s):  
Wulf-Dieter Geyer ◽  
Moshe Jarden
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Finite Fields ◽  
Galois Extension ◽  
Prime Ideals ◽  
Transfer Principle ◽  
Global Fields ◽  
Regular Extension ◽  
Root Of Unity ◽  
Result Theorem

Abstract.We use the method of Scholz and Reichardt and a transfer principle from finite fields to pseudo finite fields in order to prove the following result. THEOREM Let G be a group of order ln, where l is a prime number. Let K0be either a finite field with |K0| > l4n+4or a pseudo finite field. Suppose that l ≠ char(K0) and that K0does not contain the root of unity ζl of order l. Let K = K0(t), with t transcendental over K0. Then K has a Galois extension L with the following properties: (a) (L/K) ≅ G; (b) L/K0is a regular extension; (c) genus(L) < ; (d) K0[t] has exactly n prime ideals which ramify in L; the degree of each of them is [K0: K0]; (e) (t)∞totally decomposes in L; (f) L = K(x), withand deg(ai(t)) < deg(a1(t)) for i = 1,…,ln.

scholarly journals An explicit Baker-type lower bound of exponential values

2015 ◽  
Vol 145 (6) ◽  
pp. 1153-1182 ◽  
Author(s):  
Anne-Maria Ernvall-Hytönen ◽  
Kalle Leppälä ◽  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Linear Form ◽  
Quadratic Field ◽  
Rational Numbers ◽  
Rational Points ◽  
Imaginary Quadratic Field ◽  
Ring Of Integers

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1, . . . , eαm,m⩾ 2, whereα0= 0,α1, . . . ,αmarem+ 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies onmare improved.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

scholarly journals CONSTRUCTION OF SELF-DUAL INTEGRAL NORMAL BASES IN ABELIAN EXTENSIONS OF FINITE AND LOCAL FIELDS

2010 ◽  
Vol 06 (07) ◽  
pp. 1565-1588 ◽  
Author(s):  
ERIK JARL PICKETT
Keyword(s):  
Galois Group ◽  
Finite Fields ◽  
Galois Extension ◽  
Valuation Ring ◽  
Finite Extension ◽  
Normal Basis ◽  
Local Fields ◽  
Abelian Extensions ◽  
Construction Of Self ◽  
Odd Degree

Let F/E be a finite Galois extension of fields with abelian Galois group Γ. A self-dual normal basis for F/E is a normal basis with the additional property that Tr F/E(g(x), h(x)) = δg, h for g, h ∈ Γ. Bayer-Fluckiger and Lenstra have shown that when char (E) ≠ 2, then F admits a self-dual normal basis if and only if [F : E] is odd. If F/E is an extension of finite fields and char (E) = 2, then F admits a self-dual normal basis if and only if the exponent of Γ is not divisible by 4. In this paper, we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let K be a finite extension of ℚp, let L/K be a finite abelian Galois extension of odd degree and let [Formula: see text] be the valuation ring of L. We define AL/K to be the unique fractional [Formula: see text]-ideal with square equal to the inverse different of L/K. It is known that a self-dual integral normal basis exists for AL/K if and only if L/K is weakly ramified. Assuming p ≠ 2, we construct such bases whenever they exist.

scholarly journals MULTIPLICATIVE ORDER OF GAUSS PERIODS

2010 ◽  
Vol 06 (04) ◽  
pp. 877-882 ◽  
Author(s):  
OMRAN AHMADI ◽  
IGOR E. SHPARLINSKI ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Lower Bound ◽  
Finite Fields ◽  
Normal Bases ◽  
Multiplicative Order ◽  
Gauss Periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of von zur Gathen and Shparlinski.

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