  In this paper, the author introduces hyperfields and give some facts about
the roots and multiplicities of polynomials over hyperfield based on \cite{2}
and \cite{3}. Then he tests the sharpness of an inequality in Baker's former
work \cite{2} about the behavior of multiplicities under homomorphisms between
hyperfields and show that the inequality is not sharp under the natural
homomorphisms $\mathbb{C}\rightarrow\mathbb{P}$ and
$\mathbb{C}\rightarrow\mathbb{V}$ while it is sharp under the natural
homomorphisms $\mathbb{C}\rightarrow\mathbb{K}$,
$\mathbb{R}\rightarrow\mathbb{S}$ and $\mathbb{R}\rightarrow\mathbb{T}$
according to the previous work \cite{2} of Baker and Lorscheid.
