A note on the van der Waerden conjecture on random polynomials with symmetric Galois group for function fields
Number Theory
2022-05-26 v2 Symbolic Computation
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.
Keywords
Cite
@article{arxiv.2204.02836,
title = {A note on the van der Waerden conjecture on random polynomials with symmetric Galois group for function fields},
author = {Erich L. Kaltofen},
journal= {arXiv preprint arXiv:2204.02836},
year = {2022}
}