{"id":"e-d4f58d8e823ba8d6","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-feaf9dc130ceb19e","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":"55c43076ea006f436d303123f7eb60b17b1397dfa913a0ede6c8e54c676bcc12","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"veronese_section","certificate_raw_size":8107,"certificate_compressed_size":2284,"conclusion_name":"veronese_section_2_1 of Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-75ccd0453a9b6cec","role":"sequence","name":"Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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-feaf9dc130ceb19e","claim":"real_rooted"},{"node_id":"n-feaf9dc130ceb19e","claim":"one_signed_coefficients"},{"node_id":"n-feaf9dc130ceb19e","claim":"hadamard_kernel"},{"node_id":"n-feaf9dc130ceb19e","claim":"interlacing"}]}