{"id":"e-1689c546bc4590e7","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-4ee613bf6b082e7d","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; S_n = phi(n, D) = 1 + (1) D: Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n; the induction runs on the rows Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 2 (an upper bound of deg P_(n-L)) (exact coordinate change of the rule; the rows P_n have every root in (-inf, 0] when the rows Q_n have every root in (-inf, 0])","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"3854dd7d7578c2edb2d0b77484f8de635cbd85ff2106b549ae8da6663f27cf53","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4978,"certificate_compressed_size":1636,"conclusion_name":"Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-4ee613bf6b082e7d","claim":"one_signed_coefficients"},{"node_id":"n-4ee613bf6b082e7d","claim":"hadamard_kernel"},{"node_id":"n-4ee613bf6b082e7d","claim":"real_rooted"}]}