{"id":"e-56f9a973a14d65f2","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-3c85c2a2c3ff061d","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":"abc1c782680b030d28b6e0e18f755db0f294526846bbabb2052eae6f241fcc49","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3885,"certificate_compressed_size":1269,"conclusion_name":"Triangle read by rows: T(n,m) (0<=m<=n) = number of positive n-digit numbers that have exactly m copies of a specific, previously selected positive base-10 digit among its digits.","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-3c85c2a2c3ff061d","claim":"one_signed_coefficients"},{"node_id":"n-3c85c2a2c3ff061d","claim":"hadamard_kernel"},{"node_id":"n-3c85c2a2c3ff061d","claim":"real_rooted"},{"node_id":"n-3c85c2a2c3ff061d","claim":"interlacing"}]}