{"id":"e-2415aa95c8df0d72","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-63d7f3b1eb8cf1f2","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":"a86734e4d7c81c61df5369cfbd46f0417a3e11c03992773ec352973319639aa7","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15583,"certificate_compressed_size":2992,"conclusion_name":"shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-375ba152c4aebf61","role":"sequence","name":"A127733 * A007318 as infinite lower triangular matrices.","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-63d7f3b1eb8cf1f2","claim":"hadamard_kernel"},{"node_id":"n-63d7f3b1eb8cf1f2","claim":"real_rooted"},{"node_id":"n-63d7f3b1eb8cf1f2","claim":"one_signed_coefficients"}]}