Let G be a finite cyclic group. Every sequence S over G can be written in the form $S=\left(n₁g\right)·...·\left(n_{l}g\right)$ where g ∈ G and $n₁,...,n_{l}i\in [1,ord\left(g\right)]$, and the index ind(S) is defined to be the minimum of $(n₁+\cdots +n_l)/ord\left(g\right)$ over all possible g ∈ G such that ⟨g⟩ = G. A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd(|G|,6) = 1 has index 1. This conjecture was confirmed recently for the case when |G| is a product of at most two prime powers. However, the general case is still open. In this paper, we make some...

Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of $\mathrm{Gal}(\overline{E}/E)$ satisfying a selfduality condition “of type $\mathrm{U}\left(3\right)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb{Q}]$. As important steps, and in any rank, we prove that any first order...

This paper is concerned with the estimation of the number of integer solutions to exponential equations in several variables, over function fields. We develop a method which sometimes allows to replace known exponential bounds with polynomial ones. More generally, we prove a counting result (Thm. 1) on the integer points where given exponential terms become linearly dependent over the constant field. Several applications are given to equations (Cor. 1) and to the estimation of the number of equal...

Let $k\ge 2$ and define $F^{\left(k\right)}:={\left(F_n^{\left(k\right)}\right)}_{n\ge 0}$, the $k$-generalized Fibonacci sequence whose terms satisfy the recurrence relation $F_n^{\left(k\right)}=F_{n-1}^{\left(k\right)}+F_{n-2}^{\left(k\right)}+\cdots +F_{n-k}^{\left(k\right)}$, with initial conditions $0,0,\cdots ,0,1$ ($k$ terms) and such that the first nonzero term is $F_1^{\left(k\right)}=1$. The sequences $F:=F^{\left(2\right)}$ and $T:=F^{\left(3\right)}$ are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation $F_n^{\left(k\right)}=F_m^{\left(\ell \right)}$. In this note, we use transcendental tools to provide a general method for finding the intersections $F^{\left(k\right)}\cap F^{\left(m\right)}$ which gives evidence supporting...