Bracket Polynomial Expression of Discriminant-Resultants as SL2-invariant

The paper defines the discriminant–resultants DR_{n,r} via the generating series res(f_n, x∂_x f_n + t x y f_{n-2})/(a_0 a_n) = ∑_{r=0}^n DR_{n,r} t^r and proves the bracket polynomial expression (with the exceptional case DR_{2,2} = f_0^2) by an induction on n that uses the case of a common linear factor and the irreducibility of the resultant (Theorem 1.4/2.1) . The candidate solution gives a clean, direct proof: symbolically factor f_n and f_{n-2}, use multiplicativity of the resultant and the evaluation identity Res(α_0 x − α_1 y, G) = G(α_1, α_0) to obtain a product formula S(t) = ∏_j (A_j + t B_j), and then read off DR_{n,r} as the bracket-sum; it correctly treats the exceptional (2,2) case. One minor nit: in fixing the leading t^n coefficient, homogeneity in the second argument (not multiplicativity) justifies that the top t-coefficient equals Res(f_n, x y f_{n-2}); this does not affect correctness. Both arguments agree on the statement and are logically sound.