The exact multiplicity of the Wilson supercongruence in F_q[t]
Source: hexagonmath.org/abs/2611.00076v1
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…
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