{"id":"e-5342624c2722e8d8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b1a7e7698038fd9e","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":"2bafd0b330c5a973950215909e69e63674ff1db403c75df293d9b9434b244b31","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":18121,"certificate_compressed_size":3160,"conclusion_name":"shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a2d7d802498a9d22","role":"sequence","name":"Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 1)"}],"cited_by":[{"node_id":"n-b1a7e7698038fd9e","claim":"real_rooted"},{"node_id":"n-b1a7e7698038fd9e","claim":"one_signed_coefficients"},{"node_id":"n-b1a7e7698038fd9e","claim":"hadamard_kernel"},{"node_id":"n-b1a7e7698038fd9e","claim":"interlacing"}]}