Proof step
Affine, reversal and Möbius substitutions (elementary)
Image under a real-rootedness preserver affine, shift 0
- Concludes
- Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
- Premises
- shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
- shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
- Theorem
- polytool ?
- Lean
- RealRooted
Polynomial.Splits.comp_of_degree_le_one - Checked
- polytool replayed · Lean not attempted
Technical details
- Step id
e-19536f9f85e22b7b· JSON- Operation
- Image under a real-rootedness preserver
{ "index_shift": 0, "operation": "preserver_image", "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}" } - Claims
{ "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "-1" } } }- Certificate
preserver_image, SHA-25672c2526f54dafdb34f3c3a6a781d68937e24859e6ba1bc4bee7f8cdc3cd57965, 27,206 bytes (4,758 compressed); downloadpolytool replayed the certificate when the step was written.- Cited by
n-ac04e52510a48bb5(one_signed_coefficients)n-ac04e52510a48bb5(hadamard_kernel)n-ac04e52510a48bb5(real_rooted)