Proof step

Rolle’s theorem (elementary)

Image under a real-rootedness preserver derivative of power multiple, shift 1

Concludes

Premises

Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T…
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted RealRooted.DegreeDropReversal.splits_X_pow_mul_iff, Polynomial.Splits.iterate_derivative

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-6c02d6528613c109 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 1,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_of_power_multiple\",\"power\":1,\"derivative_order\":2}"
}

Claims

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

{
  "claim": "interlacing",
  "from_index": 1,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 eb49d03fb6c05a0186b2a48c0d6460f5af35afd6e365e59bd2632ef61716ba23, 13,210 bytes (3,118 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-2c8bb380cc75972b (real_rooted)
n-2c8bb380cc75972b (one_signed_coefficients)
n-2c8bb380cc75972b (hadamard_kernel)
n-2c8bb380cc75972b (interlacing)