Towards a generalization of the van der Waerden's conjecture for Sn-polynomials with integral coefficients over a fixed number field extension

TL;DR

This paper extends the van der Waerden's conjecture to polynomials over algebraic integer rings of fixed number field extensions, providing new results for specific degrees and field extensions.

Contribution

It generalizes the conjecture's validity from rational coefficients to algebraic integers in fixed number fields for certain degrees and extensions.

Findings

Established asymptotic count for polynomials over algebraic integers

Extended known cases of the conjecture to new degrees and fields

Provided bounds on the number of such polynomials with full symmetric Galois group

Abstract

The van der Waerden's Conjecture states that the set $\mathscr{P}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f/\mathbb{Q}}$ of the splitting field $K_f/\mathbb{Q}$ is the full symmetric group, has order $|\mathscr{P}_{n,N}^0(\mathbb{Q})|=(2N)^n+O_n(N^{n-1})$ as $N\rightarrow+\infty$. The conjecture has been shown previously for cubic and quartics polynomials by van der Waerden, Chow and Dietmann. Subsequently, Bhargava proved it for $n\ge6$. In this paper, we generalize the result for polynomials with coefficients in the ring of algebraic integers $\mathcal{O}_K$ of a fixed finite extension $K/\mathbb{Q}$ of degree $d$, for some values of $n$ and $d$.