{"id":"e-8ba085c2fb60bb1b","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-1faae3cb309663bf","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":"f166c32bb2b6c667b0cd16a0eef741f1ce9ff421e9c191f1f06baa7763e02ce1","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":8047,"certificate_compressed_size":2306,"conclusion_name":"veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","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-79986ab5023c2764","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","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-1faae3cb309663bf","claim":"real_rooted"},{"node_id":"n-1faae3cb309663bf","claim":"one_signed_coefficients"},{"node_id":"n-1faae3cb309663bf","claim":"hadamard_kernel"},{"node_id":"n-1faae3cb309663bf","claim":"interlacing"}]}