{"id":"e-b1668a0bcc59a077","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-694a17a74446a807","premise_count":1,"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":"a155887ce456d55fa627917ab5ebfef9442ab7b0b6bc089b85c98e5f9cc7f998","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13239,"certificate_compressed_size":2561,"conclusion_name":"basis_map_A008277 of veronese_section_2_0 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-13ce393522545255","role":"sequence","name":"veronese_section_2_0 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-694a17a74446a807","claim":"real_rooted"},{"node_id":"n-694a17a74446a807","claim":"one_signed_coefficients"},{"node_id":"n-694a17a74446a807","claim":"hadamard_kernel"}]}