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Hilbert's Irreducibility Theorem for Linear Differential Operators

Ruyong Feng, Zewang Guo, Wei Lu

Abstract

We prove a differential analogue of Hilbert's irreducibility theorem. Let $\mathcal{L}$ be a linear differential operator with coefficients in $C(\mathbb{X})(x)$ that is irreducible over $\overline{C(\mathbb{X})}(x)$, where $\mathbb{X}$ is an irreducible affine algebraic variety over an algebraically closed field $C$ of characteristic zero. We show that the set of $c\in \mathbb{X}(C)$ such that the specialized operator $\mathcal{L}^c$ of $\mathcal{L}$ remains irreducible over $C(x)$ is Zariski dense in $\mathbb{X}(C)$.

Hilbert's Irreducibility Theorem for Linear Differential Operators

Abstract

We prove a differential analogue of Hilbert's irreducibility theorem. Let be a linear differential operator with coefficients in that is irreducible over , where is an irreducible affine algebraic variety over an algebraically closed field of characteristic zero. We show that the set of such that the specialized operator of remains irreducible over is Zariski dense in .
Paper Structure (3 sections, 5 theorems, 32 equations)

This paper contains 3 sections, 5 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Degree bound for the coefficients of irreducible divisors
  3. Proof of Theorem \ref{['thm:mainresult']}

Key Result

Theorem 1.1

Assume that ${\mathcal{L} }\in C({\mathbb{X}})(x)[\delta]$ is irreducible over $\overline{C({\mathbb{X}})}(x)$. Then there exists an ad-open subset $U$ of ${\mathbb{X}}(C)$ such that for any $c\in U$, ${\mathcal{L} }^c$ is well-defined and ${\mathcal{L} }^c$ is irreducible over $C(x)$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Example 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 3.1
  • ...and 4 more