# On Homogeneous Combinations of Linear Recurrence Sequences
## 1 Introduction  
A sequence \((G_{n})_{n\geq 0}\) is a linear recurrence sequence with coefficients \(c_{0},c_{1},\ldots,c_{k-1}\), with \(c_{0}\
eq 0\), if  
\[G_{n+k}=c_{k-1}G_{n+k-1}+\cdots+c_{1}G_{n+1}+c_{0}G_{n}, \tag{1}\]  
for all positive integer \(n\). A recurrence sequence is therefore completely determined by the _initial values_\(G_{0},\ldots,G_{k-1}\), and by the coefficients \(c_{0},c_{1},\ldots,c_{k-1}\). The integer \(k\) is called the _order_ of the linear recurrence. The _characteristic polynomial_ of the sequence \((G_{n})_{n\geq 0}\) is given by  
\[G(x)=x^{k}-c_{k-1}x^{k-1}-\cdots-c_{1}x-c_{0}=(x-r_{1})^{m_{1}}\cdots(x-r_{t}) ^{m_{t}},\]  
where \(r_{1},\ldots,r_{t}\) are called the root of the recurrence. A root \(r_{j}\) of the recurrence is called a _dominant root_ if \(|r_{j}|>|r_{i}|\), for all \(j\
eq i\in\{1,2,\ldots,t\}\). A fundamental result in the theory of recurrence sequences asserts that there exist uniquely determined non-zero polynomials \(g_{1},\ldots,g_{t}\in\mathbb{Q}(\{r_{j}\}_{j=1}^{t})[x]\), with \(\deg g_{j}\leq m_{j}-1\), for \(j=1,2,\ldots,t\), such that  
\[G_{n}=g_{1}(n)r_{1}^{n}+\cdots+g_{t}(n)r_{t}^{n}. \tag{2}\]  
For a proof of (2), see (Reference [1] Theorem C.1). Also, if \(r_{1}\) is the dominant root, then \(g_{1}(n)\) is called the dominant polynomial. The well-known Fibonacci sequence is a recurrence of order 2 given by the recursion  
\[F_{n+2}=F_{n+1}+F_{n},\text{ with }F_{0}=0\text{ and }F_{1}=1.\]  
The Fibonacci numbers are known for their amazing properties (see Reference ([2] pp. 53-56) and References [3; 4; 5; 6; 7]). For example, we have  
\[F_{n}^{2}+F_{n+1}^{2}=F_{2n+1},\text{ for all }n\geq 0. \tag{3}\]  
In 2010, [PERSON] and [PERSON] [8] searched for similar identities in higher powers. They proved that if \(F_{n}^{s}+F_{n+1}^{s}\) is a Fibonacci number for all sufficiently large \(n\), then \(s=1\) or 2 (see [9] for a generalization). In 2012, [PERSON], [PERSON] and [PERSON] [10] showed that if \((G_{m})_{m}\) is a linear recurrence sequence (under weak assumptions) and \(G_{n+1}^{s}+\cdots+G_{n+\ell}^{s}\in(G_{m})_{m}\), for infinitely many positive integers \(n\), then \(s\) is bounded by an effectively computable constant depending only on \(\ell\) and the parameters of \(G_{m}\). Note that \(P(x_{1},x_{2},\ldots,x_{\ell})=x_{1}^{s}+\cdots+x_{\ell}^{s}\) is a homogeneous polynomial. For any \(\ell\geq 2\), we want to study the Diophantine equation  
\[P(G_{n+1},\ldots,G_{n+\ell})=G_{m} \tag{4}\]  
in positive integers \(n\) and \(m\), where \(P(x_{1},\ldots,x_{\ell})\) is an homogeneous polynomial with integer coefficients. In a first view of this equation, we can use some deep techniques in number theory such as results on finiteness of non-degenerate solutions to \(S\)-unit equations (see for instance Reference [11] and the references therein) to ensure that \(m\) should be a linear function in \(n\), say \(an+b\). Now, we can apply the Skolem-Mahler-Leech theorem to ensure that if  
\[P(G_{n+1},\ldots,G_{n+\ell})-G_{an+b}=0\]  
has infinitely many solutions, then they lie (with the possible exception of finitely many of them) in a union of arithmetic progressions. The goal of this work is to use more elementary tools to prove that if Equation (4) has infinitely many solutions, then the degree of \(P\) must be bounded for an effectively computable constant. This result, in particular, generalizes the main result of Reference [8]. More precisely, we have  
Theorem 1: _Let \((G_{n})_{n}\)be an integer linear recurrence sequence having dominant root, which is multiplicatively independent with the dominant polynomial. Also, let \(P(x_{1},\ldots,x_{\ell})=Q(x_{1},\ldots,x_{\ell-1})+x_{\ell}^{s}\) be an integer homogeneous polynomial of degree \(s\) (with \(\ell>1\)). If Equation (4) has infinitely many solutions in positive integers \(m\) and \(n\), then there exists an effectively computable constant \(C\) such that \(s<C\).