Proof step e-8b34bf66d342d123

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Premises

• Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 7aa87c1aebd4ddf81942853fca7c37664a23e1e14fec112c0b770d74990280c3, 23,869 bytes (6,101 compressed); download JSON

Cited by

• n-20086f90792ae22f (one_signed_coefficients)
• n-20086f90792ae22f (hadamard_kernel)
• n-20086f90792ae22f (interlacing)
• n-20086f90792ae22f (real_rooted)