{"id":"e-40caf6cac4d83759","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-54d66668a7ba7a7d","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":"3c84d2fb0ac6eabf420a7a596c8ba3a39cb5eb5a10cb8fbc255fbff4fcf8470f","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(n, k) = binomial(n+k-1, 2*k-1).","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-976cdb6e4a693ec8","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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-54d66668a7ba7a7d","claim":"real_rooted"},{"node_id":"n-54d66668a7ba7a7d","claim":"one_signed_coefficients"},{"node_id":"n-54d66668a7ba7a7d","claim":"hadamard_kernel"},{"node_id":"n-54d66668a7ba7a7d","claim":"interlacing"}]}