{"id":"e-2338c588f96b501c","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-1bf99ad40f056d33","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":"c48bf392a2e7cfc5fcdf2703bc6d04dc46bcaa5dac7e8e7d7ddc625443f54ba0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3022,"certificate_compressed_size":1146,"conclusion_name":"basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-1bf99ad40f056d33","claim":"real_rooted"},{"node_id":"n-1bf99ad40f056d33","claim":"one_signed_coefficients"},{"node_id":"n-1bf99ad40f056d33","claim":"hadamard_kernel"},{"node_id":"n-1bf99ad40f056d33","claim":"interlacing"}]}