Diophantine approximation

This article gives an introductory survey of recent progress on Diophantine problems, especially consequences coming from Schmidt’s subspace theorem, Baker’s transcendence method and Padé approximation. We present fundamental properties around Diophantine approximation and how it yields results in number theory.

Metabelian groups: Full-rank presentations, randomness and Diophantine problems

We study metabelian groups 𝐺 given by full rank finite presentations \langle A\mid R\rangle_{\mathcal{M}} in the variety ℳ of metabelian groups. We prove that 𝐺 is a product of a free metabelian subgroup of rank \max\{0,\lvert A\rvert-\lvert R\rvert\} and a virtually abelian normal subgroup, and that if \lvert R\rvert\leq\lvert A\rvert-2, then the Diophantine problem of 𝐺 is undecidable, while it is decidable if \lvert R\rvert\geq\lvert A\rvert. We further prove that if \lvert R\rvert\leq\lvert A\rvert-1, then, in any direct decomposition of 𝐺, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

Some Diophantine Problems Related to k-Fibonacci Numbers

Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

On special extrema of polynomials with applications to Diophantine problems

We prove that apart from explicitly given cases, described in terms of Dickson polynomials, a polynomial $$f\in \mathbb {Q}[x]$$f∈Q[x] can have at most one shift $$f(x)-\lambda $$f(x)-λ$$(\lambda \in \mathbb {C})$$(λ∈C) of the form $$u(g(x))^q(h(x))^k$$u(g(x))q(h(x))k with $$u\in \mathbb {C}$$u∈C, $$g,h\in \mathbb {C}[x]$$g,h∈C[x] and either $$\deg (g)=2$$deg(g)=2, k is even, $$q=k/2$$q=k/2 or $$\deg (g)\le 1$$deg(g)≤1, $$k\ge 2$$k≥2, $$q\ge 1$$q≥1. This is shown by handling the case of two possible shifts, which was an open issue. As an application, we give a precise statement yielding a description of polynomials f having infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely many S-integral solutions when the polynomial involved is composite. In the case where there are finitely many solutions, our results yield effective bounds for them. Finally, as further applications, we give effective results for polynomial values in the solutions of Pell equations and in non-degenerate binary recurrence sequences.

Random nilpotent groups, polycyclic presentations, and Diophantine problems

We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short,