{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 1"},"hypotheses":[{"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 + x, B = n + 2 x, C = x)","name":"operator_identity","statement":"the step operator T_n of the rule equals a_n(x) S_n","status":{"kind":"verified"}},{"evidence":"P_0 has degree 0; checked exactly","name":"start_row_in_region","statement":"P_0 is nonzero and real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"delta = 1, deg P_(n-1) = n - 1: Lambda(n, deg P_(n-1)) = 1 != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of Lambda(n, deg P_(n-1)) = 1 at n = 1 + u have one sign)","name":"degree_step","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":"deg P_n = deg P_(n-1) + delta for every n >= 1: 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":"B^2 - 4 A C = n^2 >= 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of B^2 - 4 A C = n^2 at n = 1 + u have one sign)","name":"weyl_step_discriminant_nonnegative","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","2","1"]},"numerator":["0","0","1"],"range":{"first":"1","last":null}},"decision":"holds"},"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 = 2 > 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of A_1 B_lead = 2 at n = 1 + 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":["2"]},"numerator":["2"],"range":{"first":"1","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_1(n) > 0","status":{"kind":"verified"}},{"evidence":"A_0 B_1 - A_1 B_0 = n > 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of A_0 B_1 - A_1 B_0 = n at n = 1 + u have one sign)","name":"weyl_step_interlacing","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","1"]},"numerator":["0","1"],"range":{"first":"1","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 = deg B = 1, the root -A_0/A_1 of A is left of the root -B_0/B_1 of B: A_0 B_1 - A_1 B_0 > 0 (with A_1 B_1 > 0)","status":{"kind":"verified"}},{"evidence":"T(x^k) has leading coefficient A_1(n), nonzero by (ii)","name":"weyl_step_leading_coefficients_one_signed","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","status":{"kind":"verified"}},{"evidence":"the coefficients free of n are >= 0 (exact); 2 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 >= 1, so the operator maps polynomials with coefficients of one sign to such polynomials","status":{"kind":"verified"}},{"evidence":"the coefficient A_0(n) = n >= 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of the coefficient A_0(n) = n at n = 1 + 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":["1","1"]},"numerator":["0","1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"every coefficient of A + B D + C D^2 is >= 0 for every n >= 1, so the operator maps polynomials with coefficients of one sign to such polynomials","status":{"kind":"verified"}},{"evidence":"the coefficient B_0(n) = n >= 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of the coefficient B_0(n) = n at n = 1 + u have one sign)","name":"weyl_step_coefficients_one_signed_B_0","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","1"]},"numerator":["0","1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"every coefficient of A + B D + C D^2 is >= 0 for every n >= 1, 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 >= 1","status":{"kind":"verified"}}],"iterated_preserver":{"degree_step":1,"induction_start":0,"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":["1"],"shape":"weyl_operator","terms":[{"coeff":{"coeffs":[["0","1"],["1","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","2"],["1","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":2}]},"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 >= 1, with a_n(x) = (1) ((1)) and S_n = A + B D + C D^2 with A = n + x, B = n + 2 x, C = x","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"],["1","0"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":2,"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 + x, B = n + 2 x, C = x: 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"}}