{"id":"e-16d16a1fc5b32d4a","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-08f49539099949d5","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; here S_n is the identity","lean_route":null,"lean_declarations":[],"certificate_hash":"c9ad9759a182389b527f08cceb1ce6b16b00d7562ab9813cb4bb95802ef61918","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":2928,"certificate_compressed_size":1104,"conclusion_name":"Even-numbered rows of Pascal's triangle.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-08f49539099949d5","claim":"real_rooted"},{"node_id":"n-08f49539099949d5","claim":"one_signed_coefficients"},{"node_id":"n-08f49539099949d5","claim":"hadamard_kernel"}]}