{"id":"e-380132feec912c84","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-4e0b19699f4cf0db","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; with lag L = 2, the induction runs in each residue class of n modulo L from its start row","lean_route":null,"lean_declarations":[],"certificate_hash":"9f38e271224a97f1eafb0245c6028baba9869495aea00cc3a644d1bd2bebf460","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3840,"certificate_compressed_size":1235,"conclusion_name":"Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 <= k <= n-1.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-4e0b19699f4cf0db","claim":"real_rooted"},{"node_id":"n-4e0b19699f4cf0db","claim":"one_signed_coefficients"},{"node_id":"n-4e0b19699f4cf0db","claim":"hadamard_kernel"}]}