Proof step e-742e5241e9a4e627

preserver_image affine, shift 0 → concludes about basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). 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\":\"-7\"}"
}

Premises

• Triangle read by rows: T(n,k) = binomial(k,n-k). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "lower": { "closed": false, "value": "3" }, "upper": { "closed": true, "value": "7" } } }
• { "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 7ec91e4a88324300dfa447af30a88295c389b903a4f85e3c7b98c4acd6e585c8, 19,444 bytes (3,792 compressed); download JSON

Cited by

• n-cc7e0c23bafd90c3 (real_rooted)
• n-cc7e0c23bafd90c3 (one_signed_coefficients)
• n-cc7e0c23bafd90c3 (interlacing)