{"chains":[{"edge":{"id":"e-ba9336e6375434bf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-7f8a59ba04fc7524","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"de60b31b3759e16a9b08e0fecb5a2a4eb9f35a359b82abe8d88664f2781a3d88","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2de752bc9bc14cd4","role":"sequence","name":"shift_x_plus_one of Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"3"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 3)"}]},"premises":[{"premise":{"position":0,"node_id":"n-2de752bc9bc14cd4","role":"sequence","name":"shift_x_plus_one of Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"3"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 3)"},"chains":[{"edge":{"id":"e-7ddc15d309ad2a75","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2de752bc9bc14cd4","premise_count":1,"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","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"3daff7e5b2fdadecd801855b63777a5bb1642e0ca5e64619982562ccb1902f3d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"3"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f7e7f08472ec987a","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":false,"value":"4"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 4)"}]},"premises":[{"premise":{"position":0,"node_id":"n-f7e7f08472ec987a","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":false,"value":"4"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 4)"},"chains":[{"edge":{"id":"e-8de75f6dc51d73d9","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-f7e7f08472ec987a","premise_count":0,"theorem":"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. 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