{"id":"e-5b79b5b4d4f1c577","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-08f49539099949d5","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":"2a396086b003b92c3cd0ac59655e19daaf8ee741212c4de518f2916e0b9cad3b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":4893,"certificate_compressed_size":1534,"conclusion_name":"Even-numbered rows of Pascal's triangle.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-cdb0db4ad473ee28","role":"sequence","name":"basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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-08f49539099949d5","claim":"one_signed_coefficients"},{"node_id":"n-08f49539099949d5","claim":"hadamard_kernel"},{"node_id":"n-08f49539099949d5","claim":"real_rooted"}]}