{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","18"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-3"],["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013610 of narayana"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-8","12","0","0"],["-32","-126","0","-54"],["-40","-354","-522","-243"],["-16","-288","-810","-540"],["0","-72","-360","-378"],["0","0","-36","-81"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","8","204","288","216"],["0","-32","186","1440","810"],["0","-88","-534","306","999"],["0","-48","-720","-1638","-540"],["0","0","-216","-1008","-918"],["0","0","0","-108","-243"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","32","222","360","270"],["0","0","48","579","1800","1161"],["0","0","0","306","1818","1917"],["0","0","-16","-108","180","810"],["0","0","0","-72","-270","-108"],["0","0","0","0","-36","-81"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","8","42","72","54"],["0","0","0","16","141","360","243"],["0","0","0","8","132","468","432"],["0","0","0","0","36","198","270"],["0","0","0","0","0","18","54"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["8","-12","0"],["20","0","0"],["16","57","18"],["4","63","45"],["0","18","36"],["0","0","9"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","24","-36","0"],["0","60","0","0"],["0","48","171","54"],["0","12","189","135"],["0","0","54","108"],["0","0","0","27"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","8","-12","0"],["0","0","20","0","0"],["0","0","16","57","18"],["0","0","4","63","45"],["0","0","0","18","36"],["0","0","0","0","9"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["0","3","0"],["4","-24","9"],["4","-9","-9"],["0","18","-9"],["0","0","9"]]},"terms":[{"coeff":{"coeffs":[["0","0","36","-54"],["0","-6","126","81"],["-8","-78","-108","81"],["16","0","-270","-162"],["0","72","180","-27"],["0","0","36","81"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","3","0","-27"],["0","0","15","54","27"],["0","-8","-24","18","54"],["0","0","-36","-126","-54"],["0","0","0","-18","-54"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["8","12","6","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","18"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-3"],["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-2","0"],["-4","-9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","2","6"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","32","144","0","-432","0"],["0","0","112","624","720","-432","0"],["0","0","152","1128","2736","2376","648"],["0","0","100","1080","4032","5832","2268"],["0","0","32","573","2916","5481","3078"],["0","0","4","159","1071","2565","2025"],["0","0","0","18","180","594","648"],["0","0","0","0","9","54","81"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","96","432","0","-1296","0"],["0","0","0","336","1872","2160","-1296","0"],["0","0","0","456","3384","8208","7128","1944"],["0","0","0","300","3240","12096","17496","6804"],["0","0","0","96","1719","8748","16443","9234"],["0","0","0","12","477","3213","7695","6075"],["0","0","0","0","54","540","1782","1944"],["0","0","0","0","0","27","162","243"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","32","144","0","-432","0"],["0","0","0","0","112","624","720","-432","0"],["0","0","0","0","152","1128","2736","2376","648"],["0","0","0","0","100","1080","4032","5832","2268"],["0","0","0","0","32","573","2916","5481","3078"],["0","0","0","0","4","159","1071","2565","2025"],["0","0","0","0","0","18","180","594","648"],["0","0","0","0","0","0","9","54","81"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","4","12"],["0","4","12"],["0","1","3"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0"],["0","0","-18","-54"],["0","0","-15","-45"],["0","0","-3","-9"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","12","72","108"],["0","0","10","60","90"],["0","0","2","12","18"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","-24","-144","-216"],["0","0","-20","-120","-180"],["0","0","-4","-24","-36"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-3"],["3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","6"],["-2","-6"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1","3"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","576","3600","6048","1296"],["0","0","0","0","2496","15984","28512","9072"],["0","0","0","0","3888","25992","50976","23976"],["0","0","0","0","2808","20160","45144","29808"],["0","0","0","0","960","7857","21222","18873"],["0","0","0","0","126","1413","5076","5913"],["0","0","0","0","0","81","486","729"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","240","1584","3024","1296"],["0","0","0","0","0","1056","7200","14688","7776"],["0","0","0","0","0","1656","12024","27432","18792"],["0","0","0","0","0","1200","9648","25488","22032"],["0","0","0","0","0","411","3951","12609","13365"],["0","0","0","0","0","54","774","3186","4050"],["0","0","0","0","0","0","54","324","486"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","8","48","72"],["0","0","12","72","108"],["0","0","6","36","54"],["0","0","1","6","9"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","36","216","324"],["0","0","84","504","756"],["0","0","69","414","621"],["0","0","24","144","216"],["0","0","3","18","27"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-24","-144","-216","0"],["0","0","-56","-408","-936","-648"],["0","0","-46","-372","-990","-864"],["0","0","-16","-138","-396","-378"],["0","0","-2","-18","-54","-54"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","12","108","324","324"],["0","0","0","16","144","432","432"],["0","0","0","7","63","189","189"],["0","0","0","1","9","27","27"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","48","288","432","0"],["0","0","112","780","1656","972"],["0","0","92","696","1692","1296"],["0","0","32","255","666","567"],["0","0","4","33","90","81"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-24","-216","-648","-648"],["0","0","0","-32","-288","-864","-864"],["0","0","0","-14","-126","-378","-378"],["0","0","0","-2","-18","-54","-54"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","1","6","9"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-6","-9"],["-4","-6"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["3"],["1"]]},"deriv_order":0,"shift":2}]},"statement":"w(n) E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = f(n, theta) = 1 + (2) theta + (1) theta^2; so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 1 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"from_index":2,"justification":"the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 2; T = f(n, theta) = 1 + (2) theta + (1) theta^2 is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 2","kind":"weak","orientation":"unordered","status":"claimed"},"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 2, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 for every n >= n0","status":{"kind":"verified"}},{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":1,"input_interlacing":{"interlacing":{"from_index":2,"justification":{"premises":["the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 2; T = f(n, theta) = 1 + (2) theta + (1) theta^2 is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 2"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"proved"},"lean_replay":"not_attempted","operation":{"description":"f(n, theta) = 1 + (2) theta + (1) theta^2","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","18"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-3"],["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2, for every n >= 1"},"parent_proof":{"route":"three_term_recurrence","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every parent row used is zero or real-rooted with all roots in (-inf, 0] (coefficients of one sign)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"strict","interlacing_from":2,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 1, is nonzero and real-rooted, with simple roots from n = 1 on; P_(n-1) strictly interlaces P_n for n >= 2; every root lies in (-inf, 0)"},"hypotheses":[{"evidence":"f(n) = n + 1 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = n + 1 at n = 3 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["4","1"]},"numerator":["1","1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 3","status":{"kind":"verified"}},{"evidence":"f(n) a_n = 4 n - 2 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = 4 n - 2 at n = 3 + u have one sign)","name":"leading_coefficient_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["10","4"]},"numerator":["-2","4"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"Q_1 has degree 0, Q_2 degree 1; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_1 and Q_2 have simple real roots, degrees e and e + 1, and Q_1 strictly interlaces Q_2","status":{"kind":"verified"}},{"evidence":"beta_3 = 9/20 (exact)","name":"first_coefficient_positive","statement":"beta_3 = c_3 L_1 / (a_3 L_2) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (9/16 n^2 - 9/8 n)/(n^2 - 2 n + 3/4) > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of beta_n = (9/16 n^2 - 9/8 n)/(n^2 - 2 n + 3/4) at n = 4 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["3/4","-2","1"],"method":{"kind":"taylor_shift","shifted_denominator":["35/4","6","1"],"shifted_numerator":["9/2","27/8","9/16"]},"numerator":["0","-9/8","9/16"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 4","status":{"kind":"verified"}},{"evidence":"seeds checked exactly; positive-ratio induction at t = 0 with g(n) = (3/4 n + 3/4)/(n - 1/2) (the exact ratio r_n): g(n) > 0 for n >= 3, r_3 >= g(3) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 4 + u have one sign)","name":"root_upper_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["0","3/16","-5/4","3","-3","1"],"method":{"kind":"taylor_shift","shifted_denominator":["1715/4","10339/16","1547/4","115","17","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"every root of Q_n is < 0 for every n >= 1 (region (-inf, 0) for P)","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 3, with a_n = (4 n - 2)/(n + 1), b_n = (6 n - 3)/(n + 1), c_n = (9 n - 18)/(n + 1); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","18"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","-3"],["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"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","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero constant","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 3, with a_n = (4 n - 2)/(n + 1), b_n = (6 n - 3)/(n + 1), c_n = (9 n - 18)/(n + 1); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2): every P_n, n >= 1, is nonzero and real-rooted, with simple roots from n = 1 on; P_(n-1) strictly interlaces P_n for n >= 2; every root lies in (-inf, 0)","route":"three_term_recurrence","success":true,"variant":"three_term_recurrence"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 3, with a_n = (4 n - 2)/(n + 1), b_n = (6 n - 3)/(n + 1), c_n = (9 n - 18)/(n + 1); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","every P_n, n >= 1, is nonzero and real-rooted, with simple roots from n = 1 on; P_(n-1) strictly interlaces P_n for n >= 2; every root lies in (-inf, 0)"],"theorem":"three_term_recurrence: 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"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"three_term_recurrence","rows_checked":10,"statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2 for every n >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}