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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)"},"verdict":"proved"},"map_theorem":{"basis":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"basis_name":"A111125","conclusions":[{"hypotheses":["characteristic_parameter_positive","initial_slope_nonnegative"],"input_interval":{"lower":{"closed":true,"value":{"lower_bound":"-1","radicand":"1","rational_part":"-1","sqrt_coefficient":"0","text":"-1","upper_bound":"-1"}},"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"nonnegative_coefficients":false,"output_interval":{},"sharp":false,"statement":"every real polynomial with all roots in E = [-1, 1] has a real-rooted image"},{"hypotheses":["characteristic_parameter_positive","initial_slope_nonnegative","target_interval"],"input_interval":{"lower":{"closed":true,"value":{"lower_bound":"-1","radicand":"1","rational_part":"-1","sqrt_coefficient":"0","text":"-1","upper_bound":"-1"}},"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"nonnegative_coefficients":false,"output_interval":{"lower":{"closed":true,"value":{"lower_bound":"-4","radicand":"1","rational_part":"-4","sqrt_coefficient":"0","text":"-4","upper_bound":"-4"}},"upper":{"closed":true,"value":{"lower_bound":"0","radicand":"1","rational_part":"0","sqrt_coefficient":"0","text":"0","upper_bound":"0"}}},"sharp":false,"statement":"every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-4, 0]"},{"hypotheses":["characteristic_parameter_positive","initial_slope_nonnegative","target_interval","current_constant_nonnegative","zero_in_source","radius_positive","target_upper_end_nonpositive","initial_constant_nonnegative","constant_terms_nonnegative"],"input_interval":{"lower":{"closed":true,"value":{"lower_bound":"-1","radicand":"1","rational_part":"-1","sqrt_coefficient":"0","text":"-1","upper_bound":"-1"}},"upper":{"closed":true,"value":{"lower_bound":"0","radicand":"1","rational_part":"0","sqrt_coefficient":"0","text":"0","upper_bound":"0"}}},"nonnegative_coefficients":true,"output_interval":{"lower":{"closed":true,"value":{"lower_bound":"-4","radicand":"1","rational_part":"-4","sqrt_coefficient":"0","text":"-4","upper_bound":"-4"}},"upper":{"closed":true,"value":{"lower_bound":"0","radicand":"1","rational_part":"0","sqrt_coefficient":"0","text":"0","upper_bound":"0"}}},"sharp":false,"statement":"every PF input with all roots in [-1, 0] has a PF image; no larger [-R, 0] works"}],"hypotheses":[{"evidence":"exact","name":"characteristic_parameter_positive","statement":"K = beta^2 + a beta - c = 1 > 0","status":{"kind":"verified"}},{"evidence":"exact","name":"initial_slope_nonnegative","statement":"eta = m/d = 1 >= 0","status":{"kind":"verified"}},{"evidence":"exact","name":"target_interval","statement":"2 eta - 1 = 1 >= 0 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = 1","status":{"kind":"verified"}},{"evidence":"exact","name":"current_constant_nonnegative","statement":"a = 2 >= 0","status":{"kind":"verified"}},{"evidence":"exact","name":"zero_in_source","statement":"K >= beta^2 (1 >= 0)","status":{"kind":"verified"}},{"evidence":"exact","name":"radius_positive","statement":"R = beta + sqrt(K) = 1 > 0","status":{"kind":"verified"}},{"evidence":"exact","name":"target_upper_end_nonpositive","statement":"a + 2 beta = 2 >= 0 and a^2 + 4c = 0 >= 0","status":{"kind":"verified"}},{"evidence":"exact","name":"initial_constant_nonnegative","statement":"l = 3 >= 0","status":{"kind":"verified"}},{"evidence":"a^2 + 4c = 0 >= 0: true; 2s >= a or c >= s^2 - a s: true","name":"constant_terms_nonnegative","statement":"2l >= a or c >= l^2 - a l (with a^2 + 4c >= 0): every P_k(0) >= 0","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.orderTwoConstants_nonneg_iff"],"lean_replay":"not_attempted","schema":"polytool.recurrence-basis-map-theorem.v1","shape":{"a":"2","b":"0","c":"-1","d":"1","initial_constant":"3","initial_slope":"1","kind":"affine_current","normalization":"1"},"status":"proved","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)"},"schema":"polytool.recurrence-basis-map-lift.v1"},"kind":"basis_map"}