Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups

The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality for a generic polynomial. In this paper, the relations between the Galois group (respectively, \emph{the Galois-like groups}) and the triviality of the exponent lattice of a polynomial are investigated. The \bbbq\emph{-trivial} pairs, which are at the heart of the relations between the Galois group and the triviality of the exponent lattice of a polynomial, are characterized. An effective algorithm is developed to recognize these pairs. Based on this, a new algorithm is designed to prove the triviality of the exponent lattice of a generic irreducible polynomial, which considerably improves a state-of-the-art algorithm of the same type when the polynomial degree becomes larger. In addition, the concept of the Galois-like groups of a polynomial is introduced. Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.Comment: 19 pages,2 figure

A SURVEY ON KK-FREENESS

We say that an integer n is k–free (k 2) if for every prime p the valuation vp(n) < k. If f : N ! Z, we consider the enumerating function Sk f (x) defined as the number of positive integers n x such that f(n) is k–free. When f is the identity then Sk f (x) counts the k–free positive integers up to x. We review the history of Sk f (x) in the special cases when f is the identity, the characteristic function of an arithmetic progression a polynomial, arithmetic. In each section we present the proof of the simplest case of the problem in question using exclusively elementary or standard techniques

Enumerating permutation polynomials. II. kk-cycles with minimal degree

AbstractWe consider the function m[k](q) that counts the number of cycle permutations of a finite field Fq of fixed length k such that their permutation polynomial has the smallest possible degree. We prove the upper-bound m[k](q)⩽(k−1)!(q(q−1))/k for char(Fq)>e(k−3)/e and the lower-bound m[k](q)⩾ϕ(k)(q(q−1))/k for q≡1(modk). This is done by establishing a connection with the Fq-solutions of a system of equations Ak defined over Z. As example, we give complete formulas for m[k](q) when k=4,5 and partial formulas for k=6. Finally, we analyze the Galois structure of the algebraic set Ak

On the number of divisors of the least common multiples of shifted prime powers

In this paper, we give the order of magnitude for the summatory function of the number of divisors of the least common multiple of pi−1 for i=1,2,…,k when p≤x is prime

An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

Given Γ⊂Q∗ a multiplicative subgroup and m∈N+ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which ind p Γ = m, where ind p Γ = (p − 1)/|Γ p | and Γ p is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi–Pappalardi, Lenstra, Moree, Murata, Wagstaff, and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations