  Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over
$\mathbb{C}$, and let $Y_{\hbar}(\mathfrak{g})$ be the Yangian of
$\mathfrak{g}$. In this paper, we study the sets of poles of the rational
currents defining the action of $Y_{\hbar}(\mathfrak{g})$ on an arbitrary
finite-dimensional vector space $V$. Using a weak, rational version of Frenkel
and Hernandez' Baxter polynomiality, we obtain a uniform description of these
sets in terms of the Drinfeld polynomials encoding the composition factors of
$V$ and the inverse of the $q$-Cartan matrix of $\mathfrak{g}$. We then apply
this description to obtain a concrete set of sufficient conditions for the
cyclicity and simplicity of the tensor product of any two irreducible
representations, and to classify the finite-dimensional irreducible
representations of the Yangian double.
