{"id":"e-d4fed5f2404ee52f","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-2cabfd060b04f15d","premise_count":0,"theorem":"interval basis maps: for L f = (a + b t) f + kappa (t - l)(t - h) f', l < h, kappa > 0, b + kappa deg g > 0, and g with positive leading coefficient, real-rooted in [l, h] with endpoint multiplicities m_l, m_h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in [l, h] if and only if S lies in [a + b l - kappa (h - l) m_l, a + b h + kappa (h - l) m_h] (RealRooted intervalBasisMap_maps_realRootedIn_iff; Ma and Wang, EJC 15 (2008) R17, Theorem 2 and Corollary 3); a negative leading coefficient of g only changes the sign of T","lean_route":null,"lean_declarations":[],"certificate_hash":"7ae9064d09bc8dacac5f5a6f3fc121eb6a22fc7b32bd750a7a90f0ac7c277f60","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":2501,"certificate_compressed_size":950,"conclusion_name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","claims":[{"claim":"basis_map_preserver","input_interval":"[-1, 0]","output_interval":"[-1, 0]","statement":"source [-1, 0], target [-1, 0]"}],"premises":[],"cited_by":[{"node_id":"n-2cabfd060b04f15d","claim":"basis_map_preserver"}]}