{"id":"e-f961be0564a72cf6","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-88fcd1b546ee4111","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":"6c3641eac2044a63a09591fd12f09e128bd9bcf34cfe71dac491a58159c6f5b2","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"obreschkoff_combination","certificate_raw_size":6011,"certificate_compressed_size":1640,"conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-552ecb7379159c73","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-552ecb7379159c73","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}],"cited_by":[{"node_id":"n-88fcd1b546ee4111","claim":"real_rooted"}]}