Examples on the sharpness of an inequality about multiplicities over hyperfields

TL;DR

This paper investigates the sharpness of an inequality related to polynomial multiplicities over hyperfields, showing it is not sharp in some cases but remains sharp in others, thereby clarifying the behavior of multiplicities under various hyperfield homomorphisms.

Contribution

The paper tests and clarifies the sharpness of an inequality about polynomial multiplicities over hyperfields under different homomorphisms, extending Baker's previous work.

Findings

Inequality is not sharp under $\

\

Inequality remains sharp under certain hyperfield homomorphisms.

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.