Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

reversal of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Premises

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

Theorem

polytool ?

Lean

RealRooted route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-112c17c701d61b64 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 46421b4d20904e413a5c342ee44b34c25e56097e591acd99015a1a9f44878b5a, 21,486 bytes (5,285 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-3cba0f65e01f06f3 (real_rooted)
n-3cba0f65e01f06f3 (one_signed_coefficients)
n-3cba0f65e01f06f3 (hadamard_kernel)