{
  "schema_version": 1,
  "problem_number": "AIM-SEVERAL_COMPLEX_VARIABLES-0011",
  "title": "Lower Bounds for Positive-Power Exponents in a Hermitian Polynomial Family",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a fixed integer k >= 2, let N_k(t) be the least positive integer n for which ((1+x)^(2k)-t*x^k)^n has nonnegative coefficients. Known positivity theorems imply that N_k(t) is finite for 0 <= t < T_k = 2^(2k-1), whereas no exponent works at t = T_k. We give an explicit necessary lower bound as t approaches this boundary. Writing epsilon = T_k-t, L = log(1/epsilon), and beta_k = T_k(k-1)/(4k), we prove liminf[epsilon*N_k(T_k-epsilon)-beta_k*L-beta_k*log(L)] >= C_k, with an explicit Gamma-function constant C_k. The proof compares a Gaussian contribution to a selected coefficient with a negative contribution flat to order 2k. The estimates are uniform in t near the boundary. Equivalently, they bound the exponent needed to express a power of a known bihomogeneous Hermitian form as a holomorphic squared norm. No matching upper bound or general criterion for powers of Hermitian polynomials is asserted. This AI-assisted preprint is self-audited and unrefereed; it does not resolve the general AIM problem, and absolute priority is not certified.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CV",
    "math.CA"
  ],
  "keywords": [
    "Hermitian polynomials",
    "nonnegative coefficients",
    "positive powers",
    "Laplace asymptotics",
    "squared norms",
    "exponent bounds"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-several-complex-variables-0011/",
  "pdf_url": "https://eulersolve.org/papers/aim-several-complex-variables-0011/paper.pdf?v=03e7136d849b",
  "doi": "10.5281/zenodo.23320482",
  "zenodo_record_url": "https://zenodo.org/records/23320482",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete quantitative lower-bound theorem for one explicitly specified Hermitian polynomial family, not closure of the general AIM Mappings Problem 3.2. Qualitative positivity thresholds and existence are attributed to prior work. No matching upper bound, exact asymptotic equivalence, independent review, formal verification or certified priority is claimed. Finite exact and high-precision checks supplement, but do not replace, the proof.",
  "files": {
    "paper.pdf": {
      "sha256": "03e7136d849b451ba86da6f88387c7db15039961d94d03b2fed6a83c311cc142"
    },
    "source.zip": {
      "sha256": "2dfb78447e56541cdd30bd7275cbfc186a7ffcace669de6d2bbcae6a390ed665"
    },
    "verification_report.md": {
      "sha256": "26b7c8187c229b565a15123519246432686154464a5f240059402e5d52135752"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}