{"id":"e-2429b2dd77ec8a18","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-566074d3115e7533","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"75780db4b80bf0d8fbee6e9c6fbb2813a30574a6a6b63f6b8b8772fcd87b6573","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14852,"certificate_compressed_size":2962,"conclusion_name":"basis_map_A007318 of basis_map_A013611 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8fa8f50d30f2716b","role":"sequence","name":"basis_map_A013611 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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-566074d3115e7533","claim":"real_rooted"},{"node_id":"n-566074d3115e7533","claim":"one_signed_coefficients"},{"node_id":"n-566074d3115e7533","claim":"hadamard_kernel"},{"node_id":"n-566074d3115e7533","claim":"interlacing"}]}