{"id":"e-e562a98feae0f20b","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-8229dbdd6a4b5cd9","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":"b980abbb15ff9b9ddcc1e48de5fe5cad372462adf44d0f56d344ee12e2e1d9c5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3720,"certificate_compressed_size":1280,"conclusion_name":"Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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-8229dbdd6a4b5cd9","claim":"one_signed_coefficients"},{"node_id":"n-8229dbdd6a4b5cd9","claim":"hadamard_kernel"},{"node_id":"n-8229dbdd6a4b5cd9","claim":"interlacing"},{"node_id":"n-8229dbdd6a4b5cd9","claim":"real_rooted"}]}