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[\u22123\u22122\u221a2, \u22123+2\u221a2] (rational outer bound [-1564556493/268435456, -46056243/268435456])","region_exact":"[\u22123\u22122\u221a2, \u22123+2\u221a2]","edges":["e-54689045adf560eb"]}],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"incoming_edges":[{"id":"e-54689045adf560eb","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-de8e55fb070a3937","premise_count":1,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. 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