{"id":"e-f11d2b7815563c17","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-ecd77b030b8fab94","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":"29b4002a059a305ac22aa7cc1cb0895fec9878036f68877e5bc5d40da1c2850b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3821,"certificate_compressed_size":1294,"conclusion_name":"Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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-ecd77b030b8fab94","claim":"one_signed_coefficients"},{"node_id":"n-ecd77b030b8fab94","claim":"hadamard_kernel"},{"node_id":"n-ecd77b030b8fab94","claim":"interlacing"},{"node_id":"n-ecd77b030b8fab94","claim":"real_rooted"}]}