{"id":"e-be00c68a27d32dae","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-e2695402b3f168fd","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":"524f5ebcfdff6c388f6d0ab68c34d559aec25e0b3cf8c877246e865e293d8028","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6899,"certificate_compressed_size":1402,"conclusion_name":"basis_map_A013611 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-e2695402b3f168fd","claim":"real_rooted"},{"node_id":"n-e2695402b3f168fd","claim":"one_signed_coefficients"},{"node_id":"n-e2695402b3f168fd","claim":"hadamard_kernel"}]}