The exact multiplicity of the Wilson supercongruence in F_q[t]

Let q = p^s with p an odd prime, and let P be a monic irreducible polynomial of degree d >= 2 in F_q[t]. The Wilson analogue F_d = -1 mod P holds for the product F_d of all nonzero polynomials of degree less than d, and Thakur proved that P is a Wilson prime, i.e. F_d = -1 mod P^2, exactly when all exponents of P are congruent to 0 or 1 mod p, in which case the congruence holds modulo P^(p-1). He asked whether it can hold modulo a higher power, and what the exact multiplicities are. Let r be the largest integer such that all exponents of P are congruent to 0 or 1 modulo p^r. We prove that the exact multiplicity m = v_P(F_d + 1) is 1 if r = 0, p^r - 1 if 1 <= r <= s, and q + 1 if r > s. In particular m <= q + 1, and the multiplicity of a Wilson prime in the numerator of the sum over i < d of 1/(t^(q^i) - t) is exactly m - 1. Examples over F_3, F_5 and F_9 show that all regimes occur. A script checks the theorem on all primes of small degree and on these examples.