{"id":"e-c5bcfc8ae4223484","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-b515d243eac1bbcc","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":"641eae913cdb6c05b865aeb4b7ab4a34053ade2083d3e2727c63e0f93fe650b8","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15814,"certificate_compressed_size":3116,"conclusion_name":"shift_x_plus_one of Exponential Riordan array [1, log((1-x)/(1-2x))].","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-e41c30a965e7cb9c","role":"sequence","name":"Exponential Riordan array [1, log((1-x)/(1-2x))].","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-b515d243eac1bbcc","claim":"one_signed_coefficients"},{"node_id":"n-b515d243eac1bbcc","claim":"hadamard_kernel"},{"node_id":"n-b515d243eac1bbcc","claim":"real_rooted"},{"node_id":"n-b515d243eac1bbcc","claim":"interlacing"}]}