Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monodromy groups of polynomials of composition length 2

Angelot Behajaina, Joachim König, Danny Neftin

Abstract

We study the monodromy groups of compositions of two indecomposable polynomials. In particular, we show that such monodromy groups either fulfill a certain "largeness" property, or are in an explicit list of exceptions. Such largeness results are crucial for dealing with compositions of more than two polynomials, and consequently are expected to have a wide range of applications to problems concerning the arithmetic of polynomials. Concretely, our main result is a key ingredient in the solution of a long-standing open problem due to Davenport, Lewis and Schinzel, achieved in a companion paper.

Monodromy groups of polynomials of composition length 2

Abstract

We study the monodromy groups of compositions of two indecomposable polynomials. In particular, we show that such monodromy groups either fulfill a certain "largeness" property, or are in an explicit list of exceptions. Such largeness results are crucial for dealing with compositions of more than two polynomials, and consequently are expected to have a wide range of applications to problems concerning the arithmetic of polynomials. Concretely, our main result is a key ingredient in the solution of a long-standing open problem due to Davenport, Lewis and Schinzel, achieved in a companion paper.

Paper Structure

This paper contains 31 sections, 27 theorems, 24 equations, 1 table.

Table of Contents

  1. Introduction
  2. Beyond Theorem \ref{['thm:mainres']}
  3. Solvable case
  4. Nonsolvable case
  5. Preliminaries
  6. Setup
  7. Monodromy groups of indecomposable polynomials
  8. Kernel estimates
  9. On groups $G\le U\wr V$ with $U$ cyclic
  10. On groups $G\le U\wr V$ with $U$ non-cyclic
  11. On groups $G\le U\wr V$ with $V\in \{A_n,S_n\}$
  12. Ritt moves and direct products
  13. Topological ideas: coalescing of branch points and braid group action
  14. Solvable case: proofs of Theorems \ref{['thm:affinesho']}, \ref{['thm:affS4I']}, \ref{['thm:affS4II']} and \ref{['thm:S4S4']}
  15. Proof of Theorem \ref{['thm:affinesho']}
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $g,h\in k[X]$ be indecomposable polynomials of degree $>1$. Then $g\circ h$ has a large kernel unless one of the following cases holds: In particular, if $\mathop{\mathrm{Mon}}\nolimits(g),\mathop{\mathrm{Mon}}\nolimits(h)$ are both solvable, then $g\circ h$ either has large kernel or is linearly equivalent to a monomial or a Chebyshev polynomial, or one of only two other cases holds:

Theorems & Definitions (50)

  • Theorem 1.2
  • Theorem 2.1: Composition of ${ AGL}_1$-polynomials
  • Theorem 2.2: Composition of an $S_4$-polynomial and an ${ AGL}_1$-polynomial
  • Theorem 2.3: Composition of an ${AGL}_1$-polynomial and an $S_4$-polynomial
  • Theorem 2.4: Composition of two $S_4$-polynomials
  • Theorem 2.5: Composition of an arbitrary indecomposable and a nonsolvable
  • Theorem 2.6: Composition of a nonsolvable with an $AGL_1$-polynomial
  • Theorem 2.7: Composition of a nonsolvable with an $S_4$-polynomial
  • Proposition 3.1
  • Lemma 3.2
  • ...and 40 more