Eisenstein-Dumas criterion and the action of $2\times 2$ nonsingular triangular matrices on polynomials in one variable

TL;DR

This paper characterizes when certain polynomial transformations under $GL(2,K)$ produce Eisenstein-Dumas polynomials over valued fields, providing necessary and sufficient conditions and identifying specific orbits where such polynomials exist.

Contribution

It establishes criteria for the existence of Eisenstein-Dumas polynomials within the $GL(2,K)$ orbit of a given polynomial over valued fields, extending previous understanding of polynomial transformations.

Findings

Necessary and sufficient conditions for transformations to produce Eisenstein-Dumas polynomials.

Identification of a one-parameter subset within the orbit containing such polynomials.

Results depend on the characteristic of the residue field not dividing the degree n.

Abstract

Let $K$ be a valued field (in general $K$ is not heselian) with valuation $v$ and $A(x)\in K[x]$ be a polynomial of degree $n$. We find necessary and sufficient conditions for the existence of the elements $s,t,u\in K$, $s\neq 0\neq u$, such that at least one of the polynomials $u^nA(\frac{sx+t}{u})$, $(tx+u)^nA(\frac{sx}{tx+u})$, $(ux)^nA(\frac{tx+s}{ux})$ or $(ux+t)^nA(\frac{s}{ux+t})$ is an Eisenstein-Dumas polynomial at $v$, provided that the characteristic of the residue field of $v$ does not divide $n$. Furthermore, we show that if the orbit $A(x)GL(2,K)$ contains an Eisenstein-Dumas polynomial at $v$, then an Eisenstein-Dumas polynomial at $v$ can be found in a certain one-parameter subset of this orbit.