{"id":"e-155cf4a2764fba90","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-ded0a7d0241d1354","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":"f01978d17b83b0987b5095a1cab2aadda5eaf6e62ff9ee1739f23958aaf051e6","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":17426,"certificate_compressed_size":3458,"conclusion_name":"shift_x_plus_one of Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-2"},"upper":{"closed":false,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5b857304d1bafe7a","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0)"}],"cited_by":[{"node_id":"n-ded0a7d0241d1354","claim":"real_rooted"},{"node_id":"n-ded0a7d0241d1354","claim":"one_signed_coefficients"},{"node_id":"n-ded0a7d0241d1354","claim":"hadamard_kernel"},{"node_id":"n-ded0a7d0241d1354","claim":"interlacing"}]}