Galois groups of random polynomials over the rational function field

For a fixed prime power q and natural number d we consider a random polynomial f=xⁿ+aₙ₋₁(t)xⁿ⁻¹+…+a₁(t)x+a₀(t)∈ Fq[t][x] with aᵢ drawn uniformly and independently at random from the set of all polynomials in Fq[t] of degree ≤ d. We show that with probability tending to 1 as n→∞ the Galois group Gf of f over Fq(t) is isomorphic to Sₙ₋ₖ× C, where C is cyclic, k and $|C|$ are small quantities with a simple explicit dependence on f. As a corollary we deduce that P(Gf=Sₙ\,|\,f irreducible)→ 1 as n→∞. Thus we are able to overcome the Sₙ versus Aₙ ambiguity in the most natural small box random polynomial model over Fq[t], which has not been achieved over Z so far.