Proof step e-e4032a9809c053c3

gamma_polynomial scale 1, shift 0 → concludes about Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2). JSON

Theorem

Gamma criterion RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff ?

Operation

Gamma polynomial of a palindromic parent

{
  "index_shift": 0,
  "operation": "gamma_polynomial",
  "scale": "1"
}

Premises

• Square the entries of Pascal's triangle. (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": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: gamma_polynomial, SHA-256 a0eea3b35414b51cd38f165b17a76f3f71512dff4443c8940bc7a92c5ef8ef23, 10,536 bytes (2,336 compressed); download JSON

Cited by

• n-8bfdd58279be42d7 (real_rooted)
• n-8bfdd58279be42d7 (one_signed_coefficients)
• n-8bfdd58279be42d7 (hadamard_kernel)