{"id":"e-7b97c40e19745577","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-8d887bfc40d036b9","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":"ca4413ff99fe94b834b0ccb0346cbdfd308f65a0ac6122141eabf56cdea0a24f","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6855,"certificate_compressed_size":1385,"conclusion_name":"Cube of lower triangular normalized binomial matrix.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-af754f07daf65b5e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","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-8d887bfc40d036b9","claim":"one_signed_coefficients"},{"node_id":"n-8d887bfc40d036b9","claim":"hadamard_kernel"},{"node_id":"n-8d887bfc40d036b9","claim":"real_rooted"}]}