{"id":"e-f9379b86f8cefc86","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-5482df9a9e2607d0","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"2c7966a7fdd5e4bfbdd722d24c36ab407f9a2d57e132b873c513b8941f810e35","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"obreschkoff_combination","certificate_raw_size":13143,"certificate_compressed_size":2979,"conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-2ee8dc15b8ca5b2d","role":"sequence","name":"Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-2ee8dc15b8ca5b2d","role":"sequence","name":"Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}],"cited_by":[{"node_id":"n-5482df9a9e2607d0","claim":"real_rooted"}]}