{"id":"e-9a23cb8f488d54d7","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-1fa9c1ca735cbfa5","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":"81026bead2278b98c5e8357f993a78ed08f0ef149677aa68698e9fd62de933cd","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10248,"certificate_compressed_size":2289,"conclusion_name":"basis_map_A013609 of basis_map_A013609 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","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-d9455aa236ac0835","role":"sequence","name":"basis_map_A013609 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","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-1fa9c1ca735cbfa5","claim":"real_rooted"},{"node_id":"n-1fa9c1ca735cbfa5","claim":"one_signed_coefficients"},{"node_id":"n-1fa9c1ca735cbfa5","claim":"hadamard_kernel"},{"node_id":"n-1fa9c1ca735cbfa5","claim":"interlacing"}]}