{"id":"e-5048ce3ebd740a6f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-03ac40ba4fd62e4c","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":"74df768afc942180b736cff615f9276c1ddf676929fd6ef6b47a8ee8cee483fa","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7419,"certificate_compressed_size":1623,"conclusion_name":"basis_map_A007318 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"}}}],"premises":[{"position":0,"node_id":"n-c7809537965f400d","role":"sequence","name":"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-03ac40ba4fd62e4c","claim":"real_rooted"},{"node_id":"n-03ac40ba4fd62e4c","claim":"one_signed_coefficients"},{"node_id":"n-03ac40ba4fd62e4c","claim":"hadamard_kernel"}]}