{
  "schema_version": 1,
  "problem_number": "OWR-1787-006",
  "title": "Polynomial Multiples of MacLane's Universal Functions Need Not Be Universal: A Negative Answer to a Question of Costakis",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An entire function f is universal in the sense of MacLane if its derivatives f, f′, f″, … are dense in the space H(ℂ) of entire functions with the topology of locally uniform convergence; HC(D) denotes the set of these functions. G. Costakis proved (Oberwolfach Report 6/2008, Theorem 6) that there is a residual set G ⊂ HC(D) such that pf ∈ HC(D) for every f ∈ G and every non-zero polynomial p, and asked (Question 5 there) whether pf ∈ HC(D) for every f ∈ HC(D) and every non-constant polynomial p. We show that the answer is no. Theorem A gives an explicit f_A = Σ_m a_m z^m/m! in HC(D), built from the Taylor data of an enumeration of the polynomials with non-zero Gaussian-rational coefficients placed in sparse blocks at positions n_j ≥ 2/δ_j, where δ_j bounds the moduli of the data in block j from below. Then (z f_A)^(n)(0) = n a_{n−1} is 0 or of modulus at least 2 for every n, so no derivative of z f_A lies in the open set {h : |h(0) − 1| < 1}, and z f_A ∉ HC(D); the same f_A works for every monomial c z^d with c ≠ 0, d ≥ 1. Theorem B gives, with a ramp of large Taylor coefficients in front of each block, one f_B ∈ HC(D) with p f_B ∉ HC(D) for every non-constant polynomial p: beyond a threshold depending on p, (p f_B)^(n)(0) is 0 or of modulus at least 2, and finitely many derivatives do not affect density. Hence Costakis's Theorem 6 cannot be extended from a residual set to all of HC(D). The question and the residual result are Costakis's, and the construction is the classical block construction of MacLane-type universal functions; the only new observation is that the right-hand side of the Leibniz formula (pf)^(n)(0) = Σ_i c_i n(n−1)⋯(n−i+1) a_{n−i} for p = Σ_i c_i z^i can be controlled on such functions (with the ramps for Theorem B). Gardiner (2014) answered the analogous questions for universal Taylor series negatively. The argument is elementary; we found no printed answer to the question, but a folklore version may exist. Exact computations illustrate the constructions. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.CV",
    "math.FA"
  ],
  "keywords": [
    "MacLane universal functions",
    "universal entire functions",
    "universal with respect to derivatives",
    "hypercyclic vectors",
    "differentiation operator",
    "multiplication by polynomials",
    "linear dynamics",
    "negative answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-1787-006",
    "math.CV",
    "math.FA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1787-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-1787-006/paper.pdf?v=95dd1f9efe71",
  "doi": "10.5281/zenodo.23480071",
  "zenodo_record_url": "https://zenodo.org/records/23480071",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Question 5 of Costakis (Oberwolfach Report 6/2008) negatively: an explicit universal f with zf not universal, and one universal f with pf not universal for every non-constant polynomial p. Costakis's residual positive result is his; the block construction is classical and a folklore version may exist.",
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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."
}