{"id":"e-85cb7c7abd19f98e","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-a2d7d802498a9d22","premise_count":0,"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. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"2f85d5aec1776b399fac69ff8b62435e92eaf37f45748bd17b7ad89ee835f4c0","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":4293,"certificate_compressed_size":1257,"conclusion_name":"Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).","claims":[{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-1, 1] has a real-rooted image"},{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"[-1, 1]","statement":"every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-1, 1]"}],"premises":[],"cited_by":[{"node_id":"n-a2d7d802498a9d22","claim":"basis_map_preserver"}]}