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On the distribution of the roots of polynomial z k - z k - 1 - ⋯ - z - 1

Carlos A. Gómez, Florian Luca (2021)

Commentationes Mathematicae Universitatis Carolinae

We consider the polynomial f k ( z ) = z k - z k - 1 - ⋯ - z - 1 for k ≥ 2 which arises as the characteristic polynomial of the k -generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of f k ( z ) which lie inside the unit disk.

On the divisibility of power LCM matrices by power GCD matrices

Jian Rong Zhao, Shaofang Hong, Qunying Liao, Kar-Ping Shum (2007)

Czechoslovak Mathematical Journal

Let S = { x 1 , ⋯ , x n } be a set of n distinct positive integers and e ≥ 1 an integer. Denote the n × n power GCD (resp. power LCM) matrix on S having the e -th power of the greatest common divisor ( x i , x j ) (resp. the e -th power of the least common multiple [ x i , x j ] ) as the ( i , j ) -entry of the matrix by ( ( x i , x j ) e ) (resp. ( [ x i , x j ] e ) ) . We call the set S an odd gcd closed (resp. odd lcm closed) set if every element in S is an odd number and ( x i , x j ) ∈ S (resp. [ x i , x j ] ∈ S ) for all 1 ≤ i , j ≤ n . In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that...

On the dynamics of ϕ : x → x p + a in a local field

David Adam, Youssef Fares (2010)

Actes des rencontres du CIRM

Let K be a local field, a ∈ K and ϕ : x → x p + a where p denotes the characteristic of the residue field. We prove that the minimal subsets of the dynamical system ( K , ϕ ) are cycles and describe the cycles of this system.

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