p-adic root separation and the discriminant of integer polynomials

The paper proves that for 0 ≤ ν ≤ n−1 there are ≫ Q^{n+1 − (n+2)/n · ν} irreducible degree-n polynomials of height ≍ Q with |D(P)|_p ≪ Q^{−2ν} (Theorem 3.1, formula (3.7); proof in §9) . The candidate solution claims a strictly stronger exponent, ≫ Q^{n+1 − (n+1)/(2(n−1)) · ν}, obtained by imposing high-order p-adic Taylor-vanishing at a point and counting lattice solutions. This contradicts the best-known cubic upper bound #D3,p(Q,ν) ≪ Q^{4 − 5/3·ν + ε} for all polynomials (hence for the irreducible subset), so the candidate lower bound cannot be correct for n=3 (0 ≤ ν ≤ 2) . The paper’s argument is coherent and (modulo a minor typo where Dn,p appears in place of Dirr in (9.8)) completes the proof; the model’s counting step overestimates the number of admissible polynomials.