{"id":"e-4e4f839e32ed0b09","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-58f68e861f34b78a","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":"f38b5b1b24c2dd84d5aff55ebd4a264e384eec5aaf61ce6fb8eae9e61407f06b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":2039,"certificate_compressed_size":875,"conclusion_name":"Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 2]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 2] with m = 0, target (-inf, 0]"}],"premises":[],"cited_by":[{"node_id":"n-58f68e861f34b78a","claim":"basis_map_preserver"}]}