{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0"],["-4"],["-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1","-5","-2"],["1","10","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-1","-10","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","4","2"],["1","-7","-2"],["0","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","25","4"],["0","-1","-10","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","10","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-2","-6"],["3","-9"],["3","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","7","7","-1"],["-1","-9","9","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1","9","-9","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","-2","-20","-2","0"],["0","-1","-9","9","1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","0","0","-64","-1920","-19392","-67840","-19392","-1920","-64","0","0","0"],["0","0","0","-96","-2784","-26208","-72672","72672","26208","2784","96","0","0"],["0","0","0","-48","-1344","-11712","-23232","72672","-23232","-11712","-1344","-48","0"],["0","0","0","-8","-216","-1728","-1920","15984","-15984","1920","1728","216","8"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","-192","-5248","-42816","-48384","484544","149376","15168","512","0","0","0"],["0","0","0","-576","-15744","-129216","-168192","1220928","-365952","-187200","-21504","-768","0","0"],["0","0","0","-672","-18240","-147648","-179136","1348608","-1051968","-30528","69312","9888","384","0"],["0","0","0","-384","-10304","-81216","-78336","782592","-895104","220032","67584","-3456","-1344","-64"],["0","0","0","-108","-2856","-21708","-12960","230184","-335664","145800","8928","-10044","-1512","-60"],["0","0","0","-12","-312","-2268","-288","26856","-47952","26856","-288","-2268","-312","-12"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","64","1984","20800","71872","-67904","-521408","-153152","-15296","-512","0","0","0","0"],["0","0","0","128","4128","46272","189120","17088","-907008","439872","188736","20928","736","0","0","0"],["0","0","0","96","3216","37872","165888","28224","-915552","787680","-31104","-67104","-8880","-336","0","0"],["0","0","0","32","1112","13568","60168","-13056","-441168","567744","-146352","-44832","1912","832","40","0"],["0","0","0","4","144","1800","7760","-7236","-73440","141936","-73440","-7236","7760","1800","144","4"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-32","-1248","-18048","-112512","-226752","226752","112512","18048","1248","32","0","0","0"],["0","0","0","0","-48","-1824","-25200","-141696","-171360","680256","-171360","-141696","-25200","-1824","-48","0","0"],["0","0","0","0","-24","-888","-11688","-58248","-14832","425808","-425808","14832","58248","11688","888","24","0"],["0","0","0","0","-4","-144","-1800","-7760","7236","73440","-141936","73440","7236","-7760","-1800","-144","-4"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","-96","-2672","-22976","-42560","153088","-70688","-15680","1216","352","16","0"],["0","0","0","-240","-6520","-52968","-67584","458208","-422832","50352","38208","3408","-24","-8"],["0","0","0","-216","-5760","-44712","-36288","448848","-575424","195696","29376","-9720","-1728","-72"],["0","0","0","-84","-2208","-16524","-7200","182232","-287712","140040","3744","-10692","-1536","-60"],["0","0","0","-12","-312","-2268","-288","26856","-47952","26856","-288","-2268","-312","-12"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","32","976","10032","34560","-13632","-113376","70752","12672","-1632","-368","-16","0","0"],["0","0","0","48","1544","17152","67560","-14688","-355056","354624","-39408","-29520","-2328","64","8","0"],["0","0","0","24","816","9672","40752","-18000","-299232","425808","-141408","-25416","5808","1128","48","0"],["0","0","0","4","144","1800","7760","-7236","-73440","141936","-73440","-7236","7760","1800","144","4"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-16","-608","-8400","-47232","-57120","226752","-57120","-47232","-8400","-608","-16","0","0"],["0","0","0","0","-16","-592","-7792","-38832","-9888","283872","-283872","9888","38832","7792","592","16","0"],["0","0","0","0","-4","-144","-1800","-7760","7236","73440","-141936","73440","7236","-7760","-1800","-144","-4"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-6","6"],["-9","9"],["-3","3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["2","-10","-4","0"],["3","13","-5","1"],["1","9","-9","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-2","-20","-2","0"],["0","-1","-9","9","1"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","-2","-20","-2","0"],["0","-1","-9","9","1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","0","0","-64","-1920","-19392","-67840","-19392","-1920","-64","0","0","0"],["0","0","0","-96","-2784","-26208","-72672","72672","26208","2784","96","0","0"],["0","0","0","-48","-1344","-11712","-23232","72672","-23232","-11712","-1344","-48","0"],["0","0","0","-8","-216","-1728","-1920","15984","-15984","1920","1728","216","8"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","576","16704","157248","436032","-436032","-157248","-16704","-576","0","0","0"],["0","0","0","1920","54816","499104","1217568","-2107488","129888","180192","23136","864","0","0"],["0","0","0","2592","72720","637056","1306368","-3697344","1472544","266112","-49536","-10080","-432","0"],["0","0","0","1824","50232","421704","682656","-3124320","2262096","-143184","-145248","-6912","1080","72"],["0","0","0","708","19128","153396","176544","-1376568","1460304","-363960","-86112","13428","3000","132"],["0","0","0","144","3816","29160","19008","-304992","431568","-178416","-13824","11664","1800","72"],["0","0","0","12","312","2268","288","-26856","47952","-26856","288","2268","312","12"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-192","-6144","-69120","-302592","-290688","488448","162816","16896","576","0","0","0","0"],["0","0","0","-448","-14720","-171968","-798272","-858176","1653184","-261440","-188096","-22784","-832","0","0","0"],["0","0","0","-416","-14016","-168384","-796224","-684672","2842944","-1102272","-141504","54816","9344","384","0","0"],["0","0","0","-192","-6624","-81168","-380016","-162432","2086848","-1618848","64800","93888","4512","-720","-48","0"],["0","0","0","-44","-1552","-19264","-87344","15348","656544","-851616","224736","71484","-5776","-2336","-176","-4"],["0","0","0","-4","-144","-1800","-7760","7236","73440","-141936","73440","7236","-7760","-1800","-144","-4"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","96","3808","56704","376192","943936","36928","-73856","-15488","-1184","-32","0","0","0"],["0","0","0","0","176","6816","97392","591744","1078368","-1587264","-278688","69504","20208","1696","48","0","0"],["0","0","0","0","120","4536","62088","341640","357552","-1786320","768528","268560","-7848","-8040","-792","-24","0"],["0","0","0","0","36","1328","17384","85424","12540","-641184","709680","-93216","-84900","-7824","616","112","4"],["0","0","0","0","4","144","1800","7760","-7236","-73440","141936","-73440","-7236","7760","1800","144","4"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","288","8496","83136","267392","-7680","163680","37184","768","-288","-16","0"],["0","0","0","816","23160","208616","493760","-867040","566000","-39856","-48640","-5008","-40","8"],["0","0","0","888","24464","205752","340224","-1464144","1223328","-255504","-82560","5592","1872","88"],["0","0","0","468","12592","99996","105504","-922200","1069440","-345576","-44064","20292","3408","140"],["0","0","0","120","3168","23976","13248","-257040","383616","-172656","-8640","12312","1824","72"],["0","0","0","12","312","2268","288","-26856","47952","-26856","288","2268","312","12"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-96","-3024","-33392","-145088","-182144","-82400","-181280","-36032","-416","304","16","0","0"],["0","0","0","-176","-5736","-65920","-292776","-227424","808560","-340992","75696","44688","4088","0","-8","0"],["0","0","0","-120","-4048","-48152","-215040","-57456","1157664","-986736","115392","45288","-5328","-1400","-64","0"],["0","0","0","-36","-1256","-15368","-67928","20292","514608","-709680","219792","52068","-9672","-2632","-184","-4"],["0","0","0","-4","-144","-1800","-7760","7236","73440","-141936","73440","7236","-7760","-1800","-144","-4"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","48","1856","26448","159744","283872","-453504","-55392","29184","7152","576","16","0","0"],["0","0","0","0","64","2416","32992","180528","181248","-964128","455232","131808","-13632","-5968","-544","-16","0"],["0","0","0","0","28","1032","13488","66008","7596","-499248","567744","-88272","-65484","-3928","912","120","4"],["0","0","0","0","4","144","1800","7760","-7236","-73440","141936","-73440","-7236","7760","1800","144","4"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","-4","-40","-4","0"],["0","-2","-18","18","2"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["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 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 0); so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 2 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"lean_route":"reversed","lift_summary":{"hypotheses":[{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"x^(d_n) P(1/x), d_n = 0 + 1 (n - 0)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":0,"step":1},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 0), for every n >= 0"},"parent_proof":{"route":"liu_wang_derivative_lag","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, 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":"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"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_interlacing_report":{"conclusion":{"first_index":0,"interlacing":"strict","interlacing_from":1,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2"},"hypotheses":[{"evidence":"checked exactly","name":"seed_roots_negative","statement":"P_0 and P_1 are nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"degrees 0 and 1, leading coefficients of one sign","name":"seed_degree_step","statement":"deg P_1 = deg P_0 + 1, with leading coefficients of one sign","status":{"kind":"verified"}},{"evidence":"Bezout test, exact","name":"seed_strict_interlacing","statement":"P_0 strictly interlaces P_1 (simple real roots, no common root)","status":{"kind":"verified"}},{"evidence":"lambda_n = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of lambda_n = 1 at n = 2 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] A_n + (deg P_(n-1)) [x^2] V_n > 0 for every n >= 2 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1 and [x^0] c_2 are zero polynomials","name":"multipliers_vanish_at_zero","statement":"V_n(0) = 0 and W_n(0) = 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"A_n(0) = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of A_n(0) = 1 at n = 2 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"A_n(0) > 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"V_n = 0","name":"derivative_coefficient_nonpositive","statement":"V_n(x) <= 0 for every x < 0 and every n >= 2","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = 2 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 2 + u have one sign)","name":"lag_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"W_n(x) <= 0 for every x < 0 and every n >= 2","status":{"kind":"verified"}},{"evidence":"W_n(x) = gamma(n) w(x) / f(n) with w(x) = 2 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 2 + u have one sign)","name":"derivative_or_lag_coefficient_negative","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"for every n >= 2 and x < 0, V_n(x) < 0 or W_n(x) < 0 (V_n < 0 on (-inf, 0) for every n, or W_n < 0 there for every n)","status":{"kind":"verified"}}],"lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"lean_replay":"not_attempted","lean_route":"sign","normal_form":"P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (1) + (1) x, V_n(x) = 0, W_n(x) = (2) x","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"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)","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang_derivative_lag"},"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: support must satisfy offset = 1 + ell * t_degree; fixed_delay_predecessor_window: more than one weighted lag)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (1) + (1) x, V_n(x) = 0, W_n(x) = (2) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2","route":"liu_wang_derivative_lag","success":true,"variant":"liu_wang_derivative_lag"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (1) + (1) x, V_n(x) = 0, W_n(x) = (2) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, P_n has simple roots and P_(n-1) strictly interlaces P_n (no common root), and deg P_n = deg P_(n-1) + 1 for n >= 2"],"theorem":"liu_wang_derivative_lag: 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)"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"liu_wang_derivative_lag","rows_checked":10,"statement":"every parent row P_n, n >= 0, 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 = x^(d_n) P(1/x), d_n = 0 + 1 (n - 0) for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}