{"id":"e-5194526f846dcc83","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-d28ad9841fc73317","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":"1cf762814b14713f31072593c0cfe04898fd5d2ef7f7240e351c1aa4ac72b4da","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1865,"certificate_compressed_size":731,"conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","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, 2]","output_interval":"(-inf, 0]","statement":"an input root r <= 2 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-d28ad9841fc73317","claim":"basis_map_preserver"}]}