{"id":"e-218da5a7ff42c40d","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-3ee5a104a79bfe3e","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":"34c2b0aeae0d1b0e600fd182e27fc68a2117214849976ece0fbe492fee3dc877","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":8283,"certificate_compressed_size":2244,"conclusion_name":"veronese_section_2_1 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[{"position":0,"node_id":"n-318700a428d7b97b","role":"sequence","name":"Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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-3ee5a104a79bfe3e","claim":"interlacing"},{"node_id":"n-3ee5a104a79bfe3e","claim":"real_rooted"},{"node_id":"n-3ee5a104a79bfe3e","claim":"one_signed_coefficients"},{"node_id":"n-3ee5a104a79bfe3e","claim":"hadamard_kernel"}]}