{"id":"e-951aaafc3a0e80f1","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4ed1b008678c2412","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":"f96dfa448687a5eb093200206522a9968ffbd6fc0f7d949b4238ef5462db2f46","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":12224,"certificate_compressed_size":2416,"conclusion_name":"shift_x_plus_one of veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e93cdce9b4ab8c8d","role":"sequence","name":"veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","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-4ed1b008678c2412","claim":"one_signed_coefficients"},{"node_id":"n-4ed1b008678c2412","claim":"hadamard_kernel"},{"node_id":"n-4ed1b008678c2412","claim":"real_rooted"}]}