{"id":"e-257278eb9a50cd75","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-b4fec57cd202e0f7","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"c8a95fd6883490c42e5e6df7782d96d7ed4912c18139d644a10a3ce045987475","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":7520,"certificate_compressed_size":2104,"conclusion_name":"veronese_section_2_0 of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cf38558d99f4fe37","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","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-b4fec57cd202e0f7","claim":"real_rooted"},{"node_id":"n-b4fec57cd202e0f7","claim":"one_signed_coefficients"},{"node_id":"n-b4fec57cd202e0f7","claim":"hadamard_kernel"}]}