{"id":"e-403cd3c528a50aef","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-fc7cbd97dfcf0d00","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":"843395e0cd3a88cd6d48c9335fa93262962867e4472c009e073bf3037bc85a34","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3917,"certificate_compressed_size":1279,"conclusion_name":"Triangle read by rows: T(n, k) = binomial(n, k) if n < 3, otherwise T(n, k) = binomial(n, k) + 2*binomial(n-2, k-1).","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-fc7cbd97dfcf0d00","claim":"interlacing"},{"node_id":"n-fc7cbd97dfcf0d00","claim":"one_signed_coefficients"},{"node_id":"n-fc7cbd97dfcf0d00","claim":"hadamard_kernel"},{"node_id":"n-fc7cbd97dfcf0d00","claim":"real_rooted"}]}