{"id":"e-15636530e2574f08","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-40ba0d1f2a40a97e","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":"27dd95e0d3ee68ca2377ad94b64a40beb3c6284d7254f84ecd2ef87ced932679","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"preserver_image","certificate_raw_size":22064,"certificate_compressed_size":5258,"conclusion_name":"shift_x_plus_one of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e073e5e0b021f36a","role":"sequence","name":"Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-40ba0d1f2a40a97e","claim":"interlacing"},{"node_id":"n-40ba0d1f2a40a97e","claim":"real_rooted"},{"node_id":"n-40ba0d1f2a40a97e","claim":"one_signed_coefficients"},{"node_id":"n-40ba0d1f2a40a97e","claim":"hadamard_kernel"}]}