{"id":"e-4720724f7f8010a1","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-677d34c7af372519","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":"a2ca64371a271524183852f9e53dc239f35178b10ce2f32bc3a70089fffe4627","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":2494,"certificate_compressed_size":973,"conclusion_name":"Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.","claims":[{"claim":"basis_map_preserver","input_interval":"[0, 1]","output_interval":"[-1, 0]","statement":"source [0, 1], target [-1, 0]"}],"premises":[],"cited_by":[{"node_id":"n-677d34c7af372519","claim":"basis_map_preserver"}]}