{"id":"e-48a63012ae70a1ba","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-7466465df6cec0d5","premise_count":0,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"74637fe6074d90892c32c9e25efd03a8b4e55d1934408145727314f97887c249","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":2045,"certificate_compressed_size":869,"conclusion_name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 1]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 1] with m = 0, target (-inf, 0]"}],"premises":[],"cited_by":[{"node_id":"n-7466465df6cec0d5","claim":"basis_map_preserver"}]}