{"id":"e-8de494022f32026c","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-73e4b84e56cef63a","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":"97a76d317bd667f1915cee05c8138d34ca7d903d6b082cfac14156bcde6fbadc","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":6895,"certificate_compressed_size":1399,"conclusion_name":"shift_x_plus_one of veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-11b174a1f5e78a68","role":"sequence","name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","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-73e4b84e56cef63a","claim":"one_signed_coefficients"},{"node_id":"n-73e4b84e56cef63a","claim":"hadamard_kernel"},{"node_id":"n-73e4b84e56cef63a","claim":"real_rooted"}]}