{"id":"e-e31dcd4b09a7ff12","operation":{"index_shift":-1,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"-1\",\"intercept\":\"0\"}"},"operation_detail":"affine, shift -1","conclusion_id":"n-3de13f8c869a1c34","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":"d96f79c9a9dbedd0a250792853d426c67def1cd1938e2433679ed90b1f1125ee","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":15075,"certificate_compressed_size":3186,"conclusion_name":"Triangle of coefficients of Laguerre polynomials L_n(x) (powers of x in decreasing order).","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-47fe395e9f7d5f86","role":"sequence","name":"T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","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-3de13f8c869a1c34","claim":"real_rooted"},{"node_id":"n-3de13f8c869a1c34","claim":"one_signed_coefficients"},{"node_id":"n-3de13f8c869a1c34","claim":"interlacing"}]}