THE PRODUCT OF LINEAR FORMS OVER FUNCTION FIELDS

The PDF proves that m_n = q^{n-1} for 2≤n≤q+1 and m_n = q^n for q+2≤n≤N_q (Theorem 1.1), and that m_n equals the discriminant bound disc-m_n for 2≤n≤N_q (Theorem 1.2) . The reduction/decomposition G = A·SL_n(o; p)·Γ (Theorem 1.3) underpins universal lower bounds, including m_n ≥ q^{n-1} (Proposition 2.9) and, when n≥q+2, log_q m_n ≥ n via Proposition 2.10 . The norm-form construction (1.1) gives N(L)=1 and det(L)=|d_k|^{1/2}, implying mn ≤ disc-mn (1.2) . The discriminant–genus relation |d_k|=q^{2n+2g_k−2} (Proposition 4.4) yields disc-m_n=q^{n−1+g_min(n,q)}; g_min(n,q)=0 for 2≤n≤q+1 (Lemma 4.6) and g_min(n,q)=1 for q+2≤n≤N_q (Propositions 4.10–4.11) . The candidate solution reproduces this structure and conclusions. The only minor issue is a small misattribution: the bound log_q m_n ≥ n for n≥q+2 comes immediately from Proposition 2.10 (using r=⌈n/(q+1)⌉≥2), whereas the candidate cites Lemma 2.11/Proposition 2.12 for that specific threshold. Overall, both are correct and follow substantially the same proof strategy.