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On irreducible factors of polynomials over integers

Rishu Garg, Jitender Singh

TL;DR

The paper develops new irreducibility and factorization criteria for primitive polynomials with integer coefficients by leveraging prime-power values of $f(m)$ and $p$-adic valuations of the derivatives $s_j(m)=f^{(j)}(m)/j!$ at suitably large integers $m$, guided by Newton-polygon (Dumas) methods. It proves that under precise coprimality and valuation conditions, $f$ factors into at most $r$ irreducibles in $\mathbb{Z}[z]$, with irreducibility when $r=1$, and provides dual criteria based on the leading coefficient via $s_n(m)$. The work also establishes lower bounds on irreducible factor degrees through $\Delta_f$ and demonstrates sharpness with explicit polynomials, complemented by concrete examples and constructions. Collectively, these results extend prior irreducibility criteria (notably Girstmair and its successors) and offer practical tools for certifying irreducibility or bounding factorization in $\mathbb{Z}[z]$ using prime-factor data and derivative evaluations.

Abstract

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their higher order formal derivatives at sufficiently large integer arguments. If a lower bound for the minimum possible degree of a factor of such a polynomial is known a priori, then the integer argument becomes significantly smaller, which makes the underlying factorization result easier to apply. A result on explicit lower degree factor bound for the classes of polynomials considered in this paper is also proved via Newton polygons.

On irreducible factors of polynomials over integers

TL;DR

The paper develops new irreducibility and factorization criteria for primitive polynomials with integer coefficients by leveraging prime-power values of and -adic valuations of the derivatives at suitably large integers , guided by Newton-polygon (Dumas) methods. It proves that under precise coprimality and valuation conditions, factors into at most irreducibles in , with irreducibility when , and provides dual criteria based on the leading coefficient via . The work also establishes lower bounds on irreducible factor degrees through and demonstrates sharpness with explicit polynomials, complemented by concrete examples and constructions. Collectively, these results extend prior irreducibility criteria (notably Girstmair and its successors) and offer practical tools for certifying irreducibility or bounding factorization in using prime-factor data and derivative evaluations.

Abstract

In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their higher order formal derivatives at sufficiently large integer arguments. If a lower bound for the minimum possible degree of a factor of such a polynomial is known a priori, then the integer argument becomes significantly smaller, which makes the underlying factorization result easier to apply. A result on explicit lower degree factor bound for the classes of polynomials considered in this paper is also proved via Newton polygons.
Paper Structure (3 sections, 10 theorems, 61 equations, 2 figures)

This paper contains 3 sections, 10 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Examples
  3. Proofs

Key Result

Theorem 1

Let $f=a_0+a_1 z +\cdots+a_nz^n\in \mathbb{Z}[z]$ be a primitive polynomial with $a_0a_n\neq 0$. Suppose there exists a positive integer $m\geq h_f+2$ for which $f(m)=\pm p_1^{k_1}\cdots p_r^{k_r}$, where $p_1,\ldots, p_r$ are primes which are all distinct for $r\geq 2$. For each $i=1,\ldots, r$, le Then $f$ is a product of at most $r$ irreducible polynomials in $\mathbb{Z}[z]$. In particular, if

Figures (2)

  • Figure 1: Newton polygon $NP(f)$ of $f$ for the case when $j<n$.
  • Figure 2: Newton polygon $NP(f)$ of $f$ for the case when $j=n$.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • proof : Proof of Lemma \ref{['L:2']}
  • proof : Proof of Theorem \ref{['th:1']}
  • Lemma 7
  • proof : Proof of Lemma \ref{['L:4']}
  • ...and 9 more