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Some factorization results for bivariate polynomials

Nicolae Ciprian Bonciocat, Rishu Garg, Jitender Singh

Abstract

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the degrees of the coefficients of $f$, and on information on the factorization of the constant term and of the leading coefficient of $f$, viewed as a polynomial in $y$ with coefficients in $\mathbb{K}[x]$. In particular, we provide a generalization of the bivariate version of Perron's irreducibility criterion, and similar results for polynomials in an arbitrary number of indeterminates. The proofs use non-Archimedean absolute values, that are suitable for finding information on the location of the roots of $f$ in an algebraic closure of $\mathbb{K}(x)$.

Some factorization results for bivariate polynomials

Abstract

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials over an arbitrary field . Our results rely on information on the degrees of the coefficients of , and on information on the factorization of the constant term and of the leading coefficient of , viewed as a polynomial in with coefficients in . In particular, we provide a generalization of the bivariate version of Perron's irreducibility criterion, and similar results for polynomials in an arbitrary number of indeterminates. The proofs use non-Archimedean absolute values, that are suitable for finding information on the location of the roots of in an algebraic closure of .
Paper Structure (4 sections, 13 theorems, 46 equations)

This paper contains 4 sections, 13 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proofs of the main results
  4. Examples

Key Result

Theorem A

Let $\mathbb{K}$ be a field and $f=a_0(x)+\cdots+a_{n-1}(x)y^{n-1}+a_n(x)y^n\in \mathbb{K}[x,y]$ with $n\geq 2$, $a_0,\ldots,a_{n-1}\in \mathbb{K}[x]$, $a_n\in \mathbb{K}$, and $a_0a_n\neq 0$. If then $f$ is irreducible over $\mathbb{K}[x]$.

Theorems & Definitions (25)

  • Theorem A: Perron Perron
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • ...and 15 more