{"chains":[{"edge":{"id":"e-7b68e74705acb3be","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-e73305afb136be50","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"da00b61a59dfc9f5f337a6f2b540e32bf7602a6001a6802fce1595994b0e0ae3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-356713338f14325b","role":"sequence","name":"Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-356713338f14325b","role":"sequence","name":"Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-d4b9a03bcbc74f35","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-356713338f14325b","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":"c7fde8c2936c7826de139cb094361ebd8bf2bf17b1ab6b314b0dd3c2648c13c2","verification_polytool":"replayed","verification_lean":"not_attempted","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":[]},"premises":[]}]}]}],"truncated":false,"expanded":2}