{"id":"e-f2157635d6d5ff5b","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6014e4c463d6a6b1","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"07f3688ade9f03a94b8f3fe77f1c742628d4e1af0b725a35a7eccf24d82a8167","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10274,"certificate_compressed_size":2299,"conclusion_name":"shift_x_plus_one of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1a3a6cbc48d40fe3","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(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-6014e4c463d6a6b1","claim":"one_signed_coefficients"},{"node_id":"n-6014e4c463d6a6b1","claim":"hadamard_kernel"},{"node_id":"n-6014e4c463d6a6b1","claim":"interlacing"},{"node_id":"n-6014e4c463d6a6b1","claim":"real_rooted"}]}