On the trace of linear combination of powers of algebraic numbers

Aprameyo Pal, Veekesh Kumar, R. Thangadurai

Published 2022-02-09Version 1

In this article, we prove three main results. Let $\lambda_1, \ldots, \lambda_k$ and $\alpha_1, \ldots, \alpha_k$ be nonzero algebraic numbers for some integer $k\geq 1$ and let $L = \mathbb{Q}(\lambda_1, \ldots, \lambda_k, \alpha_1, \ldots, \alpha_k)$ be the number field. Let $K$ be the Galois closure of $L$ and let $h$ be the order of the torsion subgroup of $K^\times$. We first prove an extension of a result of B. de Smit [3] as follows: {\it Take $\lambda_i = b_i \in\mathbb{Q}$ for $i=1,\ldots, k$ such that $b_1+\cdots+b_k = n\ne 0$ and $\alpha_j$'s are some of the Galois conjugates (not necessarily distinct) of $\alpha_1$ for all $j = 2, \ldots, k$ and $d\geq 1$ is the degree of $\alpha_1$. If $\displaystyle{\mathrm{Tr}}_{L/\mathbb{Q}}(b_1\alpha_1^j+\cdots+b_k\alpha_k^j) \in \mathbb{Z}$ for all $j = 1, 2, \ldots, d+d[\log_2(nd)]+1$, then $\alpha_1$ is an algebraic integer.} We then prove a general result for the infinite version as follows. {\it Suppose ${\mathrm{Tr}}_{L/\mathbb{Q}}(\lambda_i\alpha_i^a) \ne 0$ for all integers $a\in \{0, 1, 2, \ldots, h-1\}$ and for all integers $i=1,\ldots,k$. If ${\mathrm{Tr}}_{L/\mathbb{Q}}(\lambda_1\alpha_1^n+\cdots +\lambda_k \alpha^n_k)\in\mathbb{Z}$ holds true for infinitely many natural numbers $n$, then each $\alpha_i$ is an algebraic integer for all $i=1,\ldots,k$. } Here the extra assumption is a necessary condition for $k > 1$ (See Remark 1.3). We also prove a Diophantine result which states: {\it For a given rational number $p/q$, there are at most finitely many natural numbers $n$ such that ${\mathrm{Tr}}_{L/\mathbb{Q}}(\lambda_1\alpha_1^n+\cdots +\lambda_k \alpha^n_k) = p/q$.}