Examples on the sharpness of an inequality about multiplicities over hyperfields

Abstract

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.