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

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 $s,t,u\in K$, $s\neq 0\neq u$,suchthatatleastoneofthepolynomials$u^nA(\frac{sx+t}{u})$,$(tx+u)^nA(\frac{sx}{tx+u})$,$(ux)^nA(\frac{tx+s}{ux})$or$(ux+t)^nA(\frac{s}{ux+t})$isanEisenstein-Dumaspolynomialat$v$, 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.