{"id":"e-1a33bcef7c8475f7","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-8cf5a16a6dfee934","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":"99050cf856d643310d31fa0bcac65a6bca8c95e56ba16ca0e56f1b2e47ad4fd5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":2515,"certificate_compressed_size":984,"conclusion_name":"Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","claims":[{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"[-2, 0]","statement":"source [-1, 1], target [-2, 0]"}],"premises":[],"cited_by":[{"node_id":"n-8cf5a16a6dfee934","claim":"basis_map_preserver"}]}