{"chains":[{"edge":{"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. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"0034aa3aaf19ffec5c9f82ee70e355aa21cdbe681d002fac0414c9385012b3c4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]"},"chains":[{"edge":{"id":"e-5bc6f2a826329589","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-66e2c3448245b0b7","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"6d1ba874327a5f6a8fa45ffbba4379a2f7faa50e1290804ab927297624509c3f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"premises":[]},{"edge":{"id":"e-e5f3b61b305da328","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-66e2c3448245b0b7","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"d33efe518ffa4da9037567a70fd5b4bde87d46da497d576ebad6fa9a2136a47a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c2eaf5212a4dec13","role":"sequence","name":"Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-c2eaf5212a4dec13","role":"sequence","name":"Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-d285019b29e04549","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-c2eaf5212a4dec13","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"bc61429bda893d87f190825974cd5fd85b07867a52f3871e03dc2af10cca3182","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":[]},{"edge":{"id":"e-14da04f2a2dce119","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-c2eaf5212a4dec13","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"e452a077e8c9f17eb00965490ac4667f4ab40ad2bedccfbe492abb1260af7eea","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c2eaf5212a4dec13","role":"sequence","name":"Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-c2eaf5212a4dec13","role":"sequence","name":"Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"},"chains":[{"edge":{"id":"e-d285019b29e04549","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-c2eaf5212a4dec13","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"bc61429bda893d87f190825974cd5fd85b07867a52f3871e03dc2af10cca3182","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":[]},"repeat":true,"premises":[]}]}]}]}]}]}]},{"edge":{"id":"e-a25a92238630af22","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. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"efb666dad4957774570a659137bc525bbf580142f7b40d65a75db8cf2173c34f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]"}]},"premises":[{"premise":{"position":0,"node_id":"n-66e2c3448245b0b7","role":"sequence","name":"Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]"},"chains":[{"edge":{"id":"e-5bc6f2a826329589","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-66e2c3448245b0b7","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"6d1ba874327a5f6a8fa45ffbba4379a2f7faa50e1290804ab927297624509c3f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},"repeat":true,"premises":[]},{"edge":{"id":"e-e5f3b61b305da328","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-66e2c3448245b0b7","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"d33efe518ffa4da9037567a70fd5b4bde87d46da497d576ebad6fa9a2136a47a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c2eaf5212a4dec13","role":"sequence","name":"Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]},"repeat":true,"premises":[]}]}]}],"truncated":false,"expanded":6}