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Galois groups of reciprocal polynomials and the van der Waerden-Bhargava theorem

Theresa C. Anderson, Adam Bertelli, Evan M. O'Dorney

Abstract

We study the Galois groups $G_f$ of degree $2n$ reciprocal (a.k.a. palindromic) polynomials $f$ of height at most $H$, finding that $G_f$ falls short of the maximal possible group $S_2 \wr S_n$ for a proportion of all $f$ bounded above and below by constant multiples of $H^{-1} \log H$, whether or not $f$ is required to be monic. This answers a 1998 question of Davis-Duke-Sun and extends Bhargava's 2023 resolution of van der Waerden's 1936 conjecture on the corresponding question for general polynomials. Unlike in that setting, the dominant contribution comes not from reducible polynomials but from those $f$ for which $(-1)^n f(1) f(-1)$ is a square, causing $G_f$ to lie in an index-$2$ subgroup.

Galois groups of reciprocal polynomials and the van der Waerden-Bhargava theorem

Abstract

We study the Galois groups of degree reciprocal (a.k.a. palindromic) polynomials of height at most , finding that falls short of the maximal possible group for a proportion of all bounded above and below by constant multiples of , whether or not is required to be monic. This answers a 1998 question of Davis-Duke-Sun and extends Bhargava's 2023 resolution of van der Waerden's 1936 conjecture on the corresponding question for general polynomials. Unlike in that setting, the dominant contribution comes not from reducible polynomials but from those for which is a square, causing to lie in an index- subgroup.

Paper Structure

This paper contains 17 sections, 20 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Reciprocal polynomials
  4. Maximal subgroups of S₂ ≀ Sₙ
  5. Counting G1-polynomials
  6. Remarks on the monic case
  7. Counting G2-polynomials
  8. Twisted Poisson summation
  9. The index of a polynomial over a field
  10. Case I: C < H, D > H²
  11. Case II: D < H²
  12. Case III: C > H
  13. Remarks on the monic case
  14. Counting G3-polynomials
  15. Heights
  16. ...and 2 more sections

Key Result

Theorem \oldthetheorem

Let $\mathcal{E}_n^\mathrm{monic}(H)$ be the number of separable monic reciprocal polynomials $f$ of degree $2n$ with coefficients in $[-H,H]$ whose Galois group is not $S_2 \wr S_n$. Then for each $n \geq 2$,

Theorems & Definitions (40)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 30 more