{"id":"e-13ae0d32bd0cc9b1","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-914700b1060feeb2","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":"dfa79e9accc22bdd6d7a2a28f3852240c3e2db8fe5aaa550c31ebbdd13d998ca","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3711,"certificate_compressed_size":1268,"conclusion_name":"Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","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-914700b1060feeb2","claim":"one_signed_coefficients"},{"node_id":"n-914700b1060feeb2","claim":"hadamard_kernel"},{"node_id":"n-914700b1060feeb2","claim":"interlacing"},{"node_id":"n-914700b1060feeb2","claim":"real_rooted"}]}