{"id":"e-71a8bc8d427273fb","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-f60b6f201ff3063c","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":"1faa475c2d917390ebef0891258aa6c7ab1e431a0c04b919055510a8c80394d3","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":14847,"certificate_compressed_size":2775,"conclusion_name":"basis_map_A013610 of theta_plus_one 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"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c234358d95fc28fc","role":"sequence","name":"theta_plus_one 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-f60b6f201ff3063c","claim":"real_rooted"},{"node_id":"n-f60b6f201ff3063c","claim":"one_signed_coefficients"},{"node_id":"n-f60b6f201ff3063c","claim":"hadamard_kernel"},{"node_id":"n-f60b6f201ff3063c","claim":"interlacing"}]}