### On the Diophantine equation ${G}_{n}\left(x\right)={G}_{m}\left(P\left(x\right)\right)$ for third order linear recurring sequences.

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In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable. Moreover, we study similar problems in this context as the equation ${G}_{n}\left(x\right)={G}_{m}\left(P\left(x\right)\right),(m,n)\in {\mathbb{N}}^{2}$, where $\left({G}_{n}\left(x\right)\right)$ is a linear recurring sequence of polynomials and $P\left(x\right)$ is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].

In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo $1$ and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.

We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $\ell $ of terms. Then we look at the possible decompositions $f\left(x\right)=g\left(h\right(x\left)\right)$, where $g,h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $\ell $ (and we provide explicit bounds). This supports and quantifies the intuitive...

Let G be a commutative algebraic group defined over a number field K that is disjoint over K from ${}_{a}$ and satisfies the condition of semistability. Consider a linear form l on the Lie algebra of G with algebraic coefficients and an algebraic point u in a p-adic neighbourhood of the origin with the condition that l does not vanish at u. We give a lower bound for the p-adic absolute value of l(u) which depends up to an effectively computable constant only on the height of the linear form, the height...

In this paper, we study triples $a,b$ and $c$ of distinct positive integers such that $ab+1,ac+1$ and $bc+1$ are all three members of the same binary recurrence sequence.

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