{"id":"e-bcab9aa4957901b1","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-3dd184aba99388f7","premise_count":0,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"52f46633182834fb36cac9ba7697c27b0db06953864059f27d5f61d414d679fb","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1873,"certificate_compressed_size":736,"conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 9]","output_interval":"(-inf, 0]","statement":"an input root r <= 9 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-3dd184aba99388f7","claim":"basis_map_preserver"}]}