{"id":"e-71cb1ba62d8762b5","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-26633b980c3508e8","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":"c90c49f1f8d4e660952f2a5061394e446e4f29145e2f48c4e296966582fdf14b","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1901,"certificate_compressed_size":756,"conclusion_name":"Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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, 3]","output_interval":"(-inf, 0]","statement":"an input root r <= 3 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-26633b980c3508e8","claim":"basis_map_preserver"}]}