{"id":"e-63489eb06ef98a92","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-07e2f0a7469aac14","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":"7e5e459f2119dbb79de2098ba58360000cc4927ff889f69c339761c001efe86e","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":2928,"certificate_compressed_size":1111,"conclusion_name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-07e2f0a7469aac14","claim":"real_rooted"},{"node_id":"n-07e2f0a7469aac14","claim":"one_signed_coefficients"},{"node_id":"n-07e2f0a7469aac14","claim":"hadamard_kernel"}]}