{"id":"e-d84f036c2dc882d3","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-f7ad1fd21f0cb4ed","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":"a0870f29d929f892b21b8b44359c5381787fd7dfaa9238bda4dbc93f6f4e0a53","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":8955,"certificate_compressed_size":2420,"conclusion_name":"veronese_section_2_1 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[{"position":0,"node_id":"n-14dfd91463cae288","role":"sequence","name":"Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-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-f7ad1fd21f0cb4ed","claim":"real_rooted"},{"node_id":"n-f7ad1fd21f0cb4ed","claim":"one_signed_coefficients"},{"node_id":"n-f7ad1fd21f0cb4ed","claim":"hadamard_kernel"},{"node_id":"n-f7ad1fd21f0cb4ed","claim":"interlacing"}]}