{"id":"e-45dd1d47203f5c0d","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-26633b980c3508e8","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":"b25825ea0d810310ab22915dd061fa7524f453c6892c0b3fa16c41fa3fd430d0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3052,"certificate_compressed_size":1174,"conclusion_name":"Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-26633b980c3508e8","claim":"one_signed_coefficients"},{"node_id":"n-26633b980c3508e8","claim":"hadamard_kernel"},{"node_id":"n-26633b980c3508e8","claim":"interlacing"},{"node_id":"n-26633b980c3508e8","claim":"real_rooted"}]}