{"kind":"theorem_check","report":{"conclusion":{"first_index":1,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 2"},"hypotheses":[{"evidence":"f(n) = 2 n^2 + n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = 2 n^2 + n at n = 2 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["10","9","2"]},"numerator":["0","1","2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 2","status":{"kind":"verified"}},{"evidence":"q(n) delta_S(n) sum_d c_d(n, x) D^d = f(n) (g H)(n, x) o S_num holds exactly in the Ore algebra (S_n = (A + B D + C D^2 with A = n + 2 n^2 - 3 n x + 9 n^2 x, B = -x + 4 n x - 3/2 x^2 + 9/2 n x^2, C = 0) / (2 n^2 + n))","name":"operator_identity","statement":"the step operator T_n of the rule equals a_n(x) S_n","status":{"kind":"verified"}},{"evidence":"P_1 has degree 0; checked exactly","name":"start_row_in_region","statement":"P_1 is nonzero and real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"delta = 1, deg P_(n-1) = n - 2: Lambda(n, deg P_(n-1)) = 27/2 n^2 - 27/2 n + 3 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of Lambda(n, deg P_(n-1)) = 27/2 n^2 - 27/2 n + 3 at n = 2 + u have one sign)","name":"degree_step","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["30","81/2","27/2"]},"numerator":["3","-27/2","27/2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"deg P_n = deg P_(n-1) + delta for every n >= 2: the coefficient Lambda(n, e) of x^(e + delta) in f(n) T_n(x^e) is nonzero at e = deg P_(n-1)","status":{"kind":"verified"}},{"evidence":"C = 0, so B^2 - 4 A C = B^2 >= 0","name":"weyl_step_discriminant_nonnegative","statement":"(i) B^2 - 4 A C >= 0 on the real line (the symbol A + B z + C z^2 is real-rooted in z at every real x)","status":{"kind":"verified"}},{"evidence":"A_1 B_lead = 81/2 n^3 - 27 n^2 + 9/2 n > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of A_1 B_lead = 81/2 n^3 - 27 n^2 + 9/2 n at n = 2 + u have one sign)","name":"weyl_step_leading_coefficients_same_sign","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["225","765/2","216","81/2"]},"numerator":["0","9/2","-27","81/2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"(ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1: A_1(n) B_2(n) > 0","status":{"kind":"verified"}},{"evidence":"B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = -243 n^6 + 162 n^5 + 135/4 n^4 - 81/2 n^3 + 27/4 n^2 < 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = -243 n^6 + 162 n^5 + 135/4 n^4 - 81/2 n^3 + 27/4 n^2 at n = 2 + u have one sign)","name":"weyl_step_interlacing","sign_decision":{"certificate":{"condition":"negative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["-10125","-33075","-179145/4","-64341/2","-51705/4","-2754","-243"]},"numerator":["0","0","27/4","-81/2","135/4","162","-243"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"(ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1: deg A = 1, deg B = 2, B has opposite signs at the root of A and at infinity: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) < 0","status":{"kind":"verified"}},{"evidence":"A_a B_(a+1) = 81/2 n^3 - 27 n^2 + 9/2 n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of A_a B_(a+1) = 81/2 n^3 - 27 n^2 + 9/2 n at n = 2 + u have one sign)","name":"weyl_step_leading_coefficients_one_signed","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["225","765/2","216","81/2"]},"numerator":["0","9/2","-27","81/2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"(iii) T(x^k) = A x^k + k B x^(k-1) + k (k - 1) C x^(k-2) has degree deg A + k and leading coefficients of one sign for every k >= 0: with a = deg A, A_a(n) B_(a+1)(n) >= 0, so sgn(A_a) times the leading coefficient A_a + k B_(a+1) + k (k - 1) C_(a+2) is >= |A_a| > 0","status":{"kind":"verified"}},{"evidence":"delta_S(n) = 2 n^2 + n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of delta_S(n) = 2 n^2 + n at n = 2 + u have one sign)","name":"preserver_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["10","9","2"]},"numerator":["0","1","2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"the coefficients of S_n are finite at every n >= 2 (their common denominator does not vanish)","status":{"kind":"verified"}},{"evidence":"the coefficients free of n are >= 0 (exact); 4 coefficient(s) depend on n","name":"weyl_step_coefficients_one_signed","statement":"every coefficient of A + B D + C D^2 is >= 0 for every n >= 2, so the operator maps polynomials with coefficients of one sign to such polynomials","status":{"kind":"verified"}},{"evidence":"the coefficient A_0(n) = 2 n^2 + n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient A_0(n) = 2 n^2 + n at n = 2 + u have one sign)","name":"weyl_step_coefficients_one_signed_A_0","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["10","9","2"]},"numerator":["0","1","2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"every coefficient of A + B D + C D^2 is >= 0 for every n >= 2, so the operator maps polynomials with coefficients of one sign to such polynomials","status":{"kind":"verified"}},{"evidence":"the coefficient A_1(n) = 9 n^2 - 3 n >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient A_1(n) = 9 n^2 - 3 n at n = 2 + u have one sign)","name":"weyl_step_coefficients_one_signed_A_1","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["30","33","9"]},"numerator":["0","-3","9"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"every coefficient of A + B D + C D^2 is >= 0 for every n >= 2, so the operator maps polynomials with coefficients of one sign to such polynomials","status":{"kind":"verified"}},{"evidence":"the coefficient B_1(n) = 4 n - 1 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient B_1(n) = 4 n - 1 at n = 2 + u have one sign)","name":"weyl_step_coefficients_one_signed_B_1","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["7","4"]},"numerator":["-1","4"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"every coefficient of A + B D + C D^2 is >= 0 for every n >= 2, so the operator maps polynomials with coefficients of one sign to such polynomials","status":{"kind":"verified"}},{"evidence":"the coefficient B_2(n) = 9/2 n - 3/2 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient B_2(n) = 9/2 n - 3/2 at n = 2 + u have one sign)","name":"weyl_step_coefficients_one_signed_B_2","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["15/2","9/2"]},"numerator":["-3/2","9/2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"every coefficient of A + B D + C D^2 is >= 0 for every n >= 2, so the operator maps polynomials with coefficients of one sign to such polynomials","status":{"kind":"verified"}},{"evidence":"with coefficients of one sign, S_n maps a polynomial with coefficients of one sign to one with coefficients of one sign, real-rooted by Corollary 3.5, so with every root in (-inf, 0]","name":"preserver_keeps_region","statement":"S_n maps every polynomial that is zero or real-rooted with every root in (-inf, 0] to zero or such a polynomial, for every n >= 2","status":{"kind":"verified"}}],"iterated_preserver":{"degree_step":1,"induction_start":1,"multiplier_fixed_factor":["1"],"multiplier_index_factor":[{"denominator":["1"],"numerator":["1"]}],"operator_identity":"f(n) T_n = sum_d c_d(n, x) D^d equals f(n) a_n(x) S_n with a_n(x) = (1) ((1))","preserver":{"denominator":["0","1","2"],"shape":"weyl_operator","terms":[{"coeff":{"coeffs":[["0","0"],["1","-3"],["2","9"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-1","-3/2"],["0","4","9/2"]]},"deriv_order":1}]},"region":{"upper":{"closed":true,"value":"0"}},"start_degree":0},"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) S_n(P_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = n + 2 n^2 - 3 n x + 9 n^2 x, B = -x + 4 n x - 3/2 x^2 + 9/2 n x^2, C = 0) / (2 n^2 + n)","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["1","-3"],["2","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-3/2"],["0","4","9/2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = (A + B D + C D^2 with A = n + 2 n^2 - 3 n x + 9 n^2 x, B = -x + 4 n x - 3/2 x^2 + 9/2 n x^2, C = 0) / (2 n^2 + n): Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"iterated_preserver"}}