Multiplicative relations between the roots of a polynomial in $\mathbb{Q}[x]$ have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other fields. In this paper, a sufficient condition is given for a polynomial $f\in \mathbb{Q}[x]$ to have only trivial multiplicative relations between its roots, which is a generalization of those sufficient conditions proposed in Smyth (1986), Baron et al. (1995) and Dixon (1997). Based on the new condition, a subset $\mathcal{E}\subset \mathbb{Q}[x]$ is defined and proved to be generic (i.e., the set $\mathbb{Q}[x]\backslash \mathcal{E}$ is very small). We develop an algorithm deciding whether a given polynomial $f\in \mathbb{Q}[x]$ is in $\mathcal{E}$ and returning a basis of the lattice consisting of the multiplicative relations between the roots of f whenever $f\in \mathcal{E}$. The numerical experiments show that the new algorithm is very efficient for the polynomials in $\mathcal{E}$. A large number of polynomials with much higher degrees, which were intractable before, can be handled successfully with the algorithm.

多项式的根之间的乘法关系 $\mathbb{问}[X]$在算术和代数领域已经引起了很多关注，而计算这些关系的问题对于许多其他领域的研究人员来说很有趣。本文给出了多项式的充分条件$F\in \mathbb{问}[X]$在其根之间只有微不足道的乘法关系，这是对Smyth（1986），Baron等人提出的充分条件的概括。（1995）和Dixon（1997）。根据新条件，一个子集$\mathcal{Ë}\subset \mathbb{问}[X]$被定义并证明是通用的（即，集合$\mathbb{问}[X]\backslash \mathcal{Ë}$非常小）。我们开发一种算法来确定是否给定多项式$F\in \mathbb{问}[X]$ 在 $\mathcal{Ë}$并在每当返回f的根之间的乘法关系组成的格的基础时$F\in \mathcal{Ë}$。数值实验表明，该算法对多项式的求解非常有效。$\mathcal{Ë}$。该算法可以成功处理以前难于处理的大量高阶多项式。