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

2022/04/06 by Kaltofen, Erich L.
#12E25 #12F10 #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #Symbolic Computation (cs.SC)

paper · doi:10.48550/arxiv.2204.02836

Abstract

Let f(x) = xⁿ + (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/(2q²) for n = 2.

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