{"chains":[{"edge":{"id":"e-dbb2ac730432d636","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-ec35d1145e9a1211","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":"600141b1efd009f45c2f4c601d8b482cb962823acd55f568a0040a658aa15d91","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), -1 + sqrt(2)]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has a real-rooted image"},{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), -1 + sqrt(2)]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]"},{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), 0]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every PF input with all roots in [-1 - sqrt(2), 0] has a PF image; no larger [-R, 0] works"}],"premises":[]},"premises":[]},{"edge":{"id":"e-7fe1799d89540b32","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-ec35d1145e9a1211","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"certificate_hash":"a60fab79675fca2c62db1f798256442c7c2d98c9ce7bbe291c7438d224f091fb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]}],"truncated":false,"expanded":2}