{"id":"e-39fc4711b1b8289f","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-c11cb6bca0e3e7dc","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":"f1faf92361d71951aadd468849d4bbe0bd78a182db8a8a94b40325c8e081f1b5","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map_theorem","certificate_raw_size":1895,"certificate_compressed_size":751,"conclusion_name":"Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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, 8]","output_interval":"(-inf, 0]","statement":"an input root r <= 8 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[],"cited_by":[{"node_id":"n-c11cb6bca0e3e7dc","claim":"basis_map_preserver"}]}