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Galois groups of random additive polynomials

Lior Bary-Soroker, Alexei Entin, Eilidh McKemmie

Abstract

We study the distribution of the Galois group of a random $q$-additive polynomial over a rational function field: For $q$ a power of a prime $p$, let $f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots+a_1X^q+a_0X$ be a random polynomial chosen uniformly from the set of $q$-additive polynomials of degree $n$ and height $d$, that is, the coefficients are independent uniform polynomials of degree ${\rm deg}\, a_i\leq d$. The Galois group $G_f$ is a random subgroup of ${\rm GL}_n(q)$. Our main result shows that $G_f$ is almost surely large as $d,q$ are fixed and $n\to \infty$. For example, we give necessary and sufficient conditions so that ${\rm SL}_n(q)\leq G_f$ asymptotically almost surely. Our proof uses the classification of maximal subgroups of ${\rm GL}_n(q)$. We also consider the limits: $q,n$ fixed, $d\to \infty$ and $d,n$ fixed, $q\to \infty$, which are more elementary.

Galois groups of random additive polynomials

Abstract

We study the distribution of the Galois group of a random -additive polynomial over a rational function field: For a power of a prime , let be a random polynomial chosen uniformly from the set of -additive polynomials of degree and height , that is, the coefficients are independent uniform polynomials of degree . The Galois group is a random subgroup of . Our main result shows that is almost surely large as are fixed and . For example, we give necessary and sufficient conditions so that asymptotically almost surely. Our proof uses the classification of maximal subgroups of . We also consider the limits: fixed, and fixed, , which are more elementary.
Paper Structure (19 sections, 33 theorems, 96 equations)

This paper contains 19 sections, 33 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Outline of proofs and organization of the paper
  3. Preliminaries on additive polynomials
  4. $q$-additive polynomials
  5. Additive polynomials over finite fields
  6. Additive polynomials over $\mathbb{F}_q(t)$
  7. A height inequality
  8. Irreducibility
  9. Maximal subgroups of $\mathop{\mathrm{GL}}\nolimits_n(q)$
  10. Lower right block
  11. Determinants
  12. Upper right block
  13. Upper triangular block matrix groups
  14. Proof of Proposition \ref{['prop:upper right corner']}
  15. Proof of main theorems
  16. ...and 4 more sections

Key Result

Theorem 1

Fix $d>0$ and $q$ a prime power. Let $a_0,\ldots a_{n-1}$ be independent random variables, taking values in $\mathbb{F}_q[t]_{\le d}$ uniformly. Let $f = X^{q^n} + a_{n-1} X^{q^{n-1}} + \cdots + a_0 X$ and let $G_f$ be the Galois group of $f$ over $\mathbb{F}_q(t)$. Then

Theorems & Definitions (62)

  • Theorem 1
  • Theorem 2
  • Corollary 1.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • ...and 52 more