{"id":"e-815637789b0ae583","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-a03980573c4a5b72","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":"66e3b46480565df1a2a9886ef288a61913f0be6bcc15a5d5cd1c47d09dc3c03a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6857,"certificate_compressed_size":1394,"conclusion_name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7a4a43fcd9fa4a51","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.","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-a03980573c4a5b72","claim":"one_signed_coefficients"},{"node_id":"n-a03980573c4a5b72","claim":"hadamard_kernel"},{"node_id":"n-a03980573c4a5b72","claim":"real_rooted"}]}