A generalization of Dumas irreducibility criterion

Technical Summary: A Generalization of Dumas Irreducibility Criterion

Problem Statement
The paper addresses the problem of testing the irreducibility of polynomials with integer coefficients (or coefficients in a discrete valuation domain). While classical criteria such as those by Sch¨onemann, Eisenstein, Perron, and Dumas provide sufficient conditions for irreducibility, they rely on specific constraints regarding the valuation of coefficients. Specifically, the classical Dumas criterion requires the valuation of the leading coefficient to be zero and imposes a strict lower bound on the valuation of the constant term relative to intermediate coefficients. The author investigates whether these conditions can be relaxed: specifically, what happens if the valuation of a coefficient other than the leading one is zero, or if the lower bound in the valuation inequality occurs at an index $\ell > 0$ rather than at the constant term.

Methodology
The paper employs the theory of Newton polygons over discrete valuation domains $(R, v)$. The Newton polygon $NP(f)$ of a polynomial $f = \sum a_i x^i$ is defined as the lower convex hull of the set of points $\{(i, v(a_i))\}$ in the Cartesian plane.

The core methodological tool is Theorem B (Dumas), which establishes a fundamental relationship between the factorization of a polynomial and the geometry of its Newton polygon. This theorem states that the edges of the Newton polygon of a product $f = f_1 f_2$ are formed by the union of translates of the edges of the Newton polygons of the factors, arranged to form a path with increasing slopes.

The author utilizes Lemma 7, a combinatorial result concerning the number of lattice points on a line segment connecting two points $(a_1, v(a_1))$ and $(a_2, v(a_2))$, which is given by $1 + \gcd(|a_1 - a_2|, |v(a_1) - v(a_2)|)$. This lemma is critical for determining when a segment of the Newton polygon contains no interior lattice points, a condition that forces specific structural constraints on the factors of the polynomial.

Key Contributions and Results

1. Generalized Factorization Theorem (Theorem 1):
   The paper proves a mild generalization of Dumas' criterion. It establishes that for a primitive polynomial $f \in R[x]$, if there exist indices $j$ and $\ell$ ($1 \le \ell+1 \le j \le n$) such that:

   • $v(a_j) = 0$,
   • $v(a_\ell)/(j-\ell) < v(a_i)/(j-i)$ for all $i < j, i \neq \ell$,
   • $\gcd(v(a_\ell), j-\ell) = 1$,
     then any factorization $f = f_1 f_2$ must contain a factor of degree at least $j-\ell$.
   • Significance: The classical Dumas criterion is recovered as the specific case where $j=n$ and $\ell=0$. The result also generalizes previous work found in reference [6].

2. Corollary 2 (Weakened Inequality):
   A variant is presented where the strict inequality in condition (ii) of Theorem 1 is replaced by a weaker condition: $v(a_\ell) \le \frac{j-\ell}{j} v(a_i)$. This yields the same lower bound on the degree of factors.

3. Irreducibility Criteria for Integer Polynomials (Theorem 3 & Corollaries 4, 5):
   Applying the valuation $v_p$ (p-adic valuation) to polynomials in $\mathbb{Z}[x]$, the author derives new irreducibility criteria.

   • Theorem 3: If conditions analogous to Theorem 1 hold for a prime $p$ and index $j$, and either $j=n$ or all complex zeros $\theta$ satisfy $|\theta| > d$ (where $d = |a_0|/p^{v_p(a_0)}$), then $f$ is irreducible.
   • Corollary 4: A specific case of Theorem 3 where the inequality condition is relaxed to $v_p(a_0) \le v_p(a_i)$, recovering a known result from reference [4].
   • Corollary 5: A criterion for polynomials where the constant term is $a_0 = \pm p^k d$ and coefficients follow a specific divisibility pattern involving powers of $p$.

4. Factorization Bound for Multiple Primes (Theorem 6):
   The paper extends the analysis to polynomials where the constant term has multiple distinct prime factors ($a_0 = \pm p_1^{k_1} \cdots p_r^{k_r}$). If the conditions of Theorem 1 hold for each prime $p_i$ with corresponding indices $j_i$, and all complex zeros lie outside the unit disk, the polynomial is a product of at most $r$ irreducible factors in $\mathbb{Z}[x]$.

Significance and Claims
The paper claims to provide "appealing modifications" to the hypotheses of the classical Dumas criterion, thereby expanding the class of polynomials for which irreducibility or factorization bounds can be determined.

• The author demonstrates that the lower bound on the degree of a factor provided in Theorem 1 is best possible by constructing a specific example in a discrete valuation domain where a factor of exactly degree $j-\ell$ exists.
• Similarly, the bound in Theorem 6 (at most $r$ factors) is shown to be best possible via an example where the polynomial is exactly a product of $r$ irreducible factors.
• The results unify and extend existing criteria (Sch¨onemann, Eisenstein, Dumas) and connect them with the geometric properties of Newton polygons, offering a systematic way to handle cases where the "zero valuation" occurs at non-leading coefficients.

The proofs rely strictly on the properties of discrete valuation domains and the geometric interpretation of Newton polygons, without introducing new experimental methods or unverified applications.