{"id":"e-013b61d0172203f3","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-30cb3cad216ed189","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":"f7436e3b494fe470318684a0445dc5275ff32c7ff96621167cc2366618ecad60","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":5299,"certificate_compressed_size":1355,"conclusion_name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x 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":"[-6, -2]","statement":"every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-6, -2]"},{"claim":"basis_map_preserver","input_interval":"[-1, 0]","output_interval":"[-6, -2]","statement":"every PF input with all roots in [-1, 0] has a PF image; no larger [-R, 0] works"}],"premises":[],"cited_by":[{"node_id":"n-30cb3cad216ed189","claim":"basis_map_preserver"}]}