{"chains":[{"edge":{"id":"e-c1128e7ae05d26b2","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-de5746895014fee3","premise_count":0,"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":"5c335a076676397942e56669f3a7d59d83b5521210dfea7e3d51d6c0c2d57d1a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 3]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 3] with m = 0, target (-inf, 0]"}],"premises":[]},"premises":[]},{"edge":{"id":"e-09aef78f63ec9a62","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-de5746895014fee3","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"a89043961831fd9eb15ef297396d45ba77ef1de20535d0143ff4ffaf882e8b7a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}],"truncated":false,"expanded":2}