{"id":"e-62f41b682ce1c6f7","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-b6e50241110d60d5","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":"ab646850b1e25e6dd36840e2af78dc6407b17828323643fd31d15c9c4ff7ce68","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":10652,"certificate_compressed_size":2339,"conclusion_name":"basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= 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-569a801b809b6abd","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= 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-b6e50241110d60d5","claim":"interlacing"},{"node_id":"n-b6e50241110d60d5","claim":"real_rooted"},{"node_id":"n-b6e50241110d60d5","claim":"one_signed_coefficients"},{"node_id":"n-b6e50241110d60d5","claim":"hadamard_kernel"}]}