{"id":"e-cdcdb3aca64a476d","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-4ef31ba37e5b2299","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":"bc5000885f9c1c95a620c80745c851284182134b45270b1c639429b8dd6eb1ab","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"obreschkoff_combination","certificate_raw_size":5963,"certificate_compressed_size":1622,"conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*10^(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-f45d29a8d57e6cc8","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*10^(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-f45d29a8d57e6cc8","role":"sequence","name":"Triangle read by rows: T(n, m) = S1(n, m)*10^(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-4ef31ba37e5b2299","claim":"real_rooted"}]}