{"id":"e-86ba7e1c6bc69fa7","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-5b51a182e2d7adf3","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":"0a235d1d34eec7aeacc36c10a599e5cf51bbda9e4b23524f1170fbbdae349033","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":8032,"certificate_compressed_size":2297,"conclusion_name":"veronese_section_2_1 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","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-dd4aa31f3d11d152","role":"sequence","name":"Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","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-5b51a182e2d7adf3","claim":"one_signed_coefficients"},{"node_id":"n-5b51a182e2d7adf3","claim":"hadamard_kernel"},{"node_id":"n-5b51a182e2d7adf3","claim":"interlacing"},{"node_id":"n-5b51a182e2d7adf3","claim":"real_rooted"}]}