{"id":"e-d3ff78d9b396025d","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-b438a1032ada5daf","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":"56ed2109ff3e3569f47604601f83cb359649c5094640d68ca66bc33678035794","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6899,"certificate_compressed_size":1405,"conclusion_name":"basis_map_A013609 of basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-39578f69b35e8349","role":"sequence","name":"basis_map_A007318 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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-b438a1032ada5daf","claim":"real_rooted"},{"node_id":"n-b438a1032ada5daf","claim":"one_signed_coefficients"},{"node_id":"n-b438a1032ada5daf","claim":"hadamard_kernel"}]}