On the Number of Quadratic Polynomials with a Given Portrait

The paper proves asymptotic formulas for SF,1(P,B) and SF,2(P,B) by reducing to counting points on X1(P), Sym2X1(P), and applying Franke–Manin–Tschinkel on Pn and Néron’s theorem on abelian varieties, with careful control of exceptional fibres via Aut(πP) and explicit error terms (Theorems 5.2 and 6.2/6.7) . The candidate solution reaches the same leading terms, exponents, and the same shape of error terms for all genus cases Γ0, Γ1, Γ2, and its constants match the paper’s Tamagawa-measure expressions (compare the model’s constants with Definition 5.1 and Definition 6.1) . The notable flaw in the model is Lemma A, which claims an O(1) error when passing from preimages to parameters; the paper shows this contribution is of order ErrorP,F,i(B) via exceptional curves and a precise fibre-product analysis (Lemma 5.6 for d=1 and the analysis leading to Theorem 6.7 for d=2) . This mistake does not affect the final leading terms or the stated error sizes, which the model obtains correctly by other estimates. The two approaches differ technically (Schanuel-type/Masser–Vaaler/Widmer vs. FMT/Néron/Arakelov for Sym2E), but they agree on the results (e.g., Proposition 6.3 for Sym2P1, Proposition 6.4 for Sym2E, Proposition 6.5 for genus 2) .