{"id":"e-0bd629cc7b768a78","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-3933c6a9486f6595","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"0a06103209b4eca819428c1355b42c92ae50d6036b2aebdbdea091f507318201","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":8192,"certificate_compressed_size":1732,"conclusion_name":"basis_map_A008277 of Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-fefc87d068bda822","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.","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]"}],"cited_by":[{"node_id":"n-3933c6a9486f6595","claim":"real_rooted"},{"node_id":"n-3933c6a9486f6595","claim":"one_signed_coefficients"},{"node_id":"n-3933c6a9486f6595","claim":"hadamard_kernel"}]}