{"id":"e-3b2696b69808b17f","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9a30a5b089ff9460","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":"afcab859be6df23e01ec4002a220407ecc05e1cf2cdb94e30bc5557c5f45bcf1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15197,"certificate_compressed_size":3012,"conclusion_name":"shift_x_plus_one of veronese_section_2_1 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b9bba3062ead0647","role":"sequence","name":"veronese_section_2_1 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 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-9a30a5b089ff9460","claim":"one_signed_coefficients"},{"node_id":"n-9a30a5b089ff9460","claim":"hadamard_kernel"},{"node_id":"n-9a30a5b089ff9460","claim":"real_rooted"},{"node_id":"n-9a30a5b089ff9460","claim":"interlacing"}]}