A note on the van der Waerden conjecture on random polynomials with symmetric Galois group for function fields

TL;DR

This paper investigates the probability that a random polynomial over a finite field with a specific form has the full symmetric Galois group, showing it approaches certainty as the field size grows.

Contribution

It provides probabilistic results on the Galois group of certain random polynomials over function fields, extending understanding of Galois groups in finite field contexts.

Findings

Probability of Galois group being S(n) approaches 1 as q increases.

Probability of not being S(n) is at least 1/q for n >= 3.

For n=2, the probability of not being S(2) exceeds 1/q - 1/(2q^2).

Abstract

Let f(x) = x^n + (a[n-1] t + b[n-1]) x^(n-1) + ... + (a[0] t + b[0]) be of constant degree n in x and degree <= 1 in t, where all a[i],b[i] are randomly and uniformly selected from a finite field GF(q) of q elements. Then the probability that the Galois group of f over the rational function field GF(q)(t) is the symmetric group S(n) on n elements is 1 - O(1/q). Furthermore, the probability that the Galois group of f(x) over GF(q)(t) is not S(n) is >= 1/q for n >= 3 and > 1/q - 1/(2q^2) for n = 2.