{"id":"e-ac36b7d400eb36d5","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-b5ecca697c4721f9","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":"65e7690502c4830ad0a087ba9c60706e3898c421abcc8bb2793b42a3ded96e53","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6895,"certificate_compressed_size":1399,"conclusion_name":"shift_x_plus_one of shift_x_plus_one of Riordan array (1-x,x/(1-x)).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-21d2f90cb223c8f5","role":"sequence","name":"shift_x_plus_one of Riordan array (1-x,x/(1-x)).","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-b5ecca697c4721f9","claim":"hadamard_kernel"},{"node_id":"n-b5ecca697c4721f9","claim":"real_rooted"},{"node_id":"n-b5ecca697c4721f9","claim":"one_signed_coefficients"}]}