{"id":"e-982f957c1008df8b","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-d5fb64ec7bd8ef70","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":"5c53cbcbae4adba34b89870e615f520b8a72f52354d065673f12367f986f9d46","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1999,"certificate_compressed_size":854,"conclusion_name":"Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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-d5fb64ec7bd8ef70","claim":"basis_map_preserver"}]}