{"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 >= 2"},"hypotheses":[{"evidence":"f(n) = n - 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of f(n) = n - 1 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":["1","1"]},"numerator":["-1","1"],"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 = -2 + 2 n, B = -1 + n - 10 x + 10 n x, C = -x + n x - 4 x^2 + 4 n x^2) / (n - 1))","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 1; 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":"checked exactly for P_0, ..., P_0","name":"earlier_seeds_in_region","statement":"every nonzero seed row among P_0, ..., P_0 is 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)) = 4 n^3 - 6 n^2 + 2 n != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of Lambda(n, deg P_(n-1)) = 4 n^3 - 6 n^2 + 2 n 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":["12","26","18","4"]},"numerator":["0","2","-6","4"],"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":"exact check of the fixed polynomial","name":"multiplier_fixed_factor_in_region","statement":"the multiplier factor g(x) = x is real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"q_2(n) = 68 n^2 - 136 n + 68 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of q_2(n) = 68 n^2 - 136 n + 68 at n = 2 + u have one sign)","name":"weyl_step_discriminant_leading_nonnegative","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["68","136","68"]},"numerator":["68","-136","68"],"range":{"first":"2","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): with B^2 - 4 A C = q_2 x^2 + q_1 x + q_0, q_2(n) >= 0","status":{"kind":"verified"}},{"evidence":"q_0(n) = n^2 - 2 n + 1 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of q_0(n) = n^2 - 2 n + 1 at n = 2 + u have one sign)","name":"weyl_step_discriminant_constant_nonnegative","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","2","1"]},"numerator":["1","-2","1"],"range":{"first":"2","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): q_0(n) >= 0","status":{"kind":"verified"}},{"evidence":"4 q_2 q_0 - q_1^2 = 128 n^4 - 512 n^3 + 768 n^2 - 512 n + 128 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of 4 q_2 q_0 - q_1^2 = 128 n^4 - 512 n^3 + 768 n^2 - 512 n + 128 at n = 2 + 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":["128","512","768","512","128"]},"numerator":["128","-512","768","-512","128"],"range":{"first":"2","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): 4 q_2(n) q_0(n) - q_1(n)^2 >= 0 (with q_2, q_0 >= 0, the quadratic is nonnegative on the line)","status":{"kind":"verified"}},{"evidence":"A(n) = 2 n - 2 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of A(n) = 2 n - 2 at n = 2 + u have one sign)","name":"weyl_step_constant_term_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2","2"]},"numerator":["-2","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: here deg A = 0, A(n) != 0","status":{"kind":"verified"}},{"evidence":"the leading coefficient of B = 10 n - 10 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the leading coefficient of B = 10 n - 10 at n = 2 + u have one sign)","name":"weyl_step_interlacing","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["10","10"]},"numerator":["-10","10"],"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: here deg A = 0 and deg B <= 1 (vacuous), B != 0","status":{"kind":"verified"}},{"evidence":"A_a B_(a+1) = 20 n^2 - 40 n + 20 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of A_a B_(a+1) = 20 n^2 - 40 n + 20 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":["20","40","20"]},"numerator":["20","-40","20"],"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":"A_a C_(a+2) = 8 n^2 - 16 n + 8 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of A_a C_(a+2) = 8 n^2 - 16 n + 8 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":["8","16","8"]},"numerator":["8","-16","8"],"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) C_(a+2)(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) = n - 1 != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of delta_S(n) = n - 1 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":["1","1"]},"numerator":["-1","1"],"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); 5 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 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient A_0(n) = 2 n - 2 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":["2","2"]},"numerator":["-2","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 B_0(n) = n - 1 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient B_0(n) = n - 1 at n = 2 + 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":["-1","1"],"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) = 10 n - 10 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient B_1(n) = 10 n - 10 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":["10","10"]},"numerator":["-10","10"],"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 C_1(n) = n - 1 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient C_1(n) = n - 1 at n = 2 + u have one sign)","name":"weyl_step_coefficients_one_signed_C_1","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","1"]},"numerator":["-1","1"],"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 C_2(n) = 4 n - 4 >= 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of the coefficient C_2(n) = 4 n - 4 at n = 2 + u have one sign)","name":"weyl_step_coefficients_one_signed_C_2","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["4","4"]},"numerator":["-4","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":"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":["0","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) = (x) ((1))","preserver":{"denominator":["-1","1"],"shape":"weyl_operator","terms":[{"coeff":{"coeffs":[["-2"],["2"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1","-10"],["1","10"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","-1","-4"],["0","1","4"]]},"deriv_order":2}]},"region":{"upper":{"closed":true,"value":"0"}},"start_degree":1},"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) S_n(P_(n-1)) for n >= 2, with a_n(x) = (x) ((1)) and S_n = (A + B D + C D^2 with A = -2 + 2 n, B = -1 + n - 10 x + 10 n x, C = -x + n x - 4 x^2 + 4 n x^2) / (n - 1)","recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-10"],["0","1","10"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-4"],["0","0","1","4"]]},"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 = -2 + 2 n, B = -1 + n - 10 x + 10 n x, C = -x + n x - 4 x^2 + 4 n x^2) / (n - 1): 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"}}