{"id":"e-fa986ad01f1c1b33","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-e2ca539e0aa62c2f","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":"3497ffbbd02fc783684eb096bd57c87a3bcf23bf7e87db2a576738644a074e4b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":11662,"certificate_compressed_size":2189,"conclusion_name":"basis_map_A013611 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-392f49b496f79fbf","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-e2ca539e0aa62c2f","claim":"real_rooted"},{"node_id":"n-e2ca539e0aa62c2f","claim":"one_signed_coefficients"},{"node_id":"n-e2ca539e0aa62c2f","claim":"hadamard_kernel"}]}