{"id":"e-a5a16e178035b04e","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-3823d7dcffdc2ba2","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":"30ffc544b7ea95b074a1ea6bd6f09f498a58b19bab1cc140181fc4345ba47eef","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":7517,"certificate_compressed_size":2100,"conclusion_name":"veronese_section_2_1 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-3823d7dcffdc2ba2","claim":"real_rooted"},{"node_id":"n-3823d7dcffdc2ba2","claim":"one_signed_coefficients"},{"node_id":"n-3823d7dcffdc2ba2","claim":"hadamard_kernel"}]}