{"id":"e-ae3732c3a58c2035","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-5d4d13a95dac2445","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":"a25d7462aea9fdba9659ea8d0d824254d26dcb2661400e3471d93b880a2778c8","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14059,"certificate_compressed_size":2695,"conclusion_name":"basis_map_A013609 of Triangular array:  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-5fa21c8779d2bdd1","role":"sequence","name":"Triangular array:  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-5d4d13a95dac2445","claim":"real_rooted"},{"node_id":"n-5d4d13a95dac2445","claim":"one_signed_coefficients"},{"node_id":"n-5d4d13a95dac2445","claim":"hadamard_kernel"},{"node_id":"n-5d4d13a95dac2445","claim":"interlacing"}]}