{"id":"e-a82376e9d30ba1f5","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-089a9b4f824a1c20","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":"ae9c5433817fb1c79d756e5a8e62c88bc10dc7deb28ba265cf43d16c20ec3038","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3034,"certificate_compressed_size":1152,"conclusion_name":"Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","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-089a9b4f824a1c20","claim":"hadamard_kernel"},{"node_id":"n-089a9b4f824a1c20","claim":"interlacing"},{"node_id":"n-089a9b4f824a1c20","claim":"real_rooted"},{"node_id":"n-089a9b4f824a1c20","claim":"one_signed_coefficients"}]}