{"id":"e-3b5486b1b9009268","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-aa480b4ac960c965","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":"8ab3503625893c84c20fe012ecf544cf86a6bb327d1b3b20e5aa4d9d74b0651b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3885,"certificate_compressed_size":1264,"conclusion_name":"Numbers in the (2,3)-Pascal triangle (by row).","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-aa480b4ac960c965","claim":"one_signed_coefficients"},{"node_id":"n-aa480b4ac960c965","claim":"hadamard_kernel"},{"node_id":"n-aa480b4ac960c965","claim":"real_rooted"},{"node_id":"n-aa480b4ac960c965","claim":"interlacing"}]}