{"id":"e-3832154a3e5fc513","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-99e5bf113a0a46cd","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; with lag L = 3, the induction runs in each residue class of n modulo L from its start row","lean_route":null,"lean_declarations":[],"certificate_hash":"a0b3ff67e40115c83927b189671b4d592ff8c8cd642ba5045d42cba03aad21d2","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4516,"certificate_compressed_size":1242,"conclusion_name":"Triangle read by rows, A007318 rows repeated three times.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-99e5bf113a0a46cd","claim":"real_rooted"},{"node_id":"n-99e5bf113a0a46cd","claim":"one_signed_coefficients"},{"node_id":"n-99e5bf113a0a46cd","claim":"hadamard_kernel"}]}