Generalized Dumas–Eisenstein irreducibility conjecture

Let $R$ be an integral domain. Let $n>1$ be a natural number and let $\varphi_0,\varphi_1,\ldots,\varphi_n\in R$. Let $\omega$ be a Dumas valuation on $R$, and assume

$$\omega (\varphi _n)=0,\qquad \omega (\varphi _0)\in\mathbb N,\qquad \gcd (n,\omega (\varphi _0))=1.$$

Further assume one of the following two possibilities: either $\omega$ is a Dumas valuation of the first kind and

$$n\omega (\varphi _j)\leq (n-j)\omega (\varphi _0)\quad\text{for }j=1,\ldots,n-1,$$

or $\omega$ is a Dumas valuation of the second kind and

$$n\omega (\varphi _j)\geq (n-j)\omega (\varphi _0)\quad\text{for }j=1,\ldots,n-1.$$

Generalized Dumas–Eisenstein irreducibility conjecture. The polynomial

$$f=\varphi _nX^n+\cdots +\varphi _1X+\varphi _0$$

cannot be written as a product $f=gh$ with $g,h\in R[X]$ both of degree at least one; equivalently, $f$ is irreducible.

This conjecture would provide a common irreducibility criterion encompassing the classical Eisenstein–Dumas criterion and extending it to Dumas valuations of both kinds. The source describes it as providing strong evidence for a broad generalization of established Newton-polygon-based irreducibility results; its resolution status is not supplied.