{"id":"e-50ecd620d262c41e","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-094e4df3b51dddea","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":"934ab510a7c7a0b17838e9a5619b4b69ebbf143f8721d95fed58e5a5319e2888","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"obreschkoff_combination","certificate_raw_size":12994,"certificate_compressed_size":2897,"conclusion_name":"obreschkoff_combination of T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-47fe395e9f7d5f86","role":"sequence","name":"T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","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-47fe395e9f7d5f86","role":"sequence","name":"T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","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-094e4df3b51dddea","claim":"real_rooted"}]}