{"id":"e-203a3bddd606a435","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-efc52f864f8006b6","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"70abc8d5c300ab190b6ac5cc38c7237c7e5e124fcc6d4b5210e24748ee9fc367","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":16094,"certificate_compressed_size":3147,"conclusion_name":"reversal of shift_x_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eec36ee0e876d0f2","role":"sequence","name":"shift_x_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":true,"value":"-1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, -1]"}],"cited_by":[{"node_id":"n-efc52f864f8006b6","claim":"real_rooted"},{"node_id":"n-efc52f864f8006b6","claim":"one_signed_coefficients"},{"node_id":"n-efc52f864f8006b6","claim":"hadamard_kernel"}]}