{"id":"e-af983e30b523c51d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-b92815c7f855e2d1","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":"2a110dff54ac559c62afdfb767797dbf0c95730d236992944d4d6d57182ffd31","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15715,"certificate_compressed_size":3101,"conclusion_name":"shift_x_plus_one of veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 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-29efd6ec0498dec9","role":"sequence","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 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-b92815c7f855e2d1","claim":"one_signed_coefficients"},{"node_id":"n-b92815c7f855e2d1","claim":"hadamard_kernel"},{"node_id":"n-b92815c7f855e2d1","claim":"real_rooted"},{"node_id":"n-b92815c7f855e2d1","claim":"interlacing"}]}