{"id":"e-d52b551add083e42","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-0135f5d413e6cc58","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":"8bd24a2fe4c679bb0a60e949fce76ea3e24548a9d845dce138d7dd72f9e835e9","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":13622,"certificate_compressed_size":2595,"conclusion_name":"basis_map_A007318 of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","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-9496336e2b29e28e","role":"sequence","name":"Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","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-0135f5d413e6cc58","claim":"real_rooted"},{"node_id":"n-0135f5d413e6cc58","claim":"one_signed_coefficients"},{"node_id":"n-0135f5d413e6cc58","claim":"hadamard_kernel"},{"node_id":"n-0135f5d413e6cc58","claim":"interlacing"}]}