{"id":"e-ef1c37b6ab776d5a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-95b4aa73235b5230","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"16c458f225966c623e7389e56612d02936561315bf9a59171df7b9d0bbf09f3c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":7450,"certificate_compressed_size":2172,"conclusion_name":"mobius_x_over_x_plus_one of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e65f982fa1b436c5","role":"sequence","name":"Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-95b4aa73235b5230","claim":"real_rooted"},{"node_id":"n-95b4aa73235b5230","claim":"one_signed_coefficients"},{"node_id":"n-95b4aa73235b5230","claim":"hadamard_kernel"}]}