{"id":"e-bef9620f9d798d0f","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-47fe395e9f7d5f86","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 = A + B D + C D^2 with A = n + x, B = x, C = 0: Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]; 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 - 1 (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":[],"certificate_hash":"636ba4353f9dca36252959c20440ce0982176027bf83370394c8c906d2cf260e","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":7745,"certificate_compressed_size":2051,"conclusion_name":"T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-47fe395e9f7d5f86","claim":"one_signed_coefficients"},{"node_id":"n-47fe395e9f7d5f86","claim":"hadamard_kernel"},{"node_id":"n-47fe395e9f7d5f86","claim":"real_rooted"}]}