The sharp exponent for ordinary Hadamard powers of quartics with positive coefficients

Summary

Take a degree-four polynomial whose coefficients are all positive and whose four roots are real. Now raise every coefficient to the same positive power, leaving the powers of the variable unchanged. Which exponents guarantee that the roots stay real, for every such polynomial?

This unrefereed candidate gives a sharp answer. Exponent one works because it changes nothing. Immediately above one, some polynomials lose real roots. Universal preservation returns at a threshold of about 1.147720381237014, and every larger exponent works. Above the threshold all four output roots are distinct. At the threshold, a repeated output root is possible only when all four input roots coincide.

The paper supplies a complete proposed proof, not a numerical search promoted to a universal result. Exact code checks selected algebra and the threshold's decimal enclosure. External mathematical reconstruction and specialist review remain to be obtained.

Summary for specialists

Let $\mathcal R_4$ contain degree-exactly-four polynomials $f(x)=\sum_{k=0}^4 a_kx^k$ with $a_k>0$ and only real zeros, including multiplicities. Define ordinary coefficient powers by $T_pf(x)=\sum_{k=0}^4 a_k^p x^k$. The candidate proves

where $p_*>1$ is the unique root above one of

$$6^p-2\cdot4^p+2=0.$$

For $p>p_*$ every output has four distinct negative zeros. At $p=p_*$, multiple output zeros occur exactly for $f(x)=c(x+r)^4$, $c,r>0$; the normalized output factors as

$$ (x+1)^2\bigl(x^2+(4^{p_*}-2)x+1\bigr). $$

The statement uses ordinary, not binomial-normalized, coefficients. It does not classify arbitrary degrees, zero-coefficient boundary inputs or general coefficient functions. The retained secondary AIM identifier is context; the originating page was not authenticated in the current audit.

Technical account

The difficult step is showing that a symmetric example controls an unsymmetrical, noncompact family. Reciprocal symmetry is not assumed of the input.

First, Newton inequalities provide a deliberately nonsharp successful tail. A two-sided root perturbation at a universally preserving exponent yields a linear functional annihilating the input's repeated-root factor times a specified polynomial space. This restricts which multiplicities can occur when output roots collide.

Next, strict lower-degree preservation excludes failures whose input roots separate into widely different scales. A last-failing-exponent argument then forces any hypothetical failure above the proposed threshold to produce a compact boundary input at a universal preserver. It does not assume that the preservation set is connected or monotone.

For quartics, the boundary functional excludes all but multiplicity patterns $(4)$ and $(2,2)$. Those remaining inputs become reciprocal after positive rescaling. A reciprocal quartic reduces to a quadratic in $x+x^{-1}$, where two explicit inequalities yield the sharp threshold. Closedness is invoked only after preservation above the threshold has been established; a separate boundary argument proves the equality classification.

Evidence, assurance and limitations

The evidence is the complete written proof, a dependency map, exact auxiliary algebra, rational threshold bounds, package-integrity controls, source audits and a response to the supplied AI-assisted review. Publication checks include fresh-extraction replay, optimized-mode refusal, PDF inspection and Linux CI. Their dated receipts identify the exact scope and bytes checked.

The algebra checker does not prove the scale-escape or compactness lemmas. Deliberately corrupted expressions are local controls, not whole-proof mutation tests. The assertion-based algebra checker refuses optimized Python; the separate package verifier uses explicit errors and continues to enforce hashes.

The five-role editorial process is producer-coordinated internal AI review. It is not unaffiliated reproduction, formal verification, authenticated specialist review or journal peer review. Some later full texts and the original AIM page remained inaccessible. No absolute novelty, historical priority, four-star grade or measured workflow impact is claimed.

Relationship to earlier work

Wang and Zhang's 2013 paper already establishes cubic preservation, gives a quartic counterexample, and supplies a degree-dependent sufficient tail. Białas and Białas-Cież's 2017 comment concerns Hurwitz stability; the 2024 paper by Białas, Białas-Cież and Kudra also exhibits quartic failure at exponent 1.147. These are prior results, not new discoveries in this release.

The proposed contribution is the exact universal quartic endpoint and its equality classification. Finite-free multiplicative convolution uses a different binomial normalization. Kudra's 2026 idealizer preprint concerns fixed linear multipliers preserving the left half-plane. Neither comparison by itself establishes or refutes this nonlinear real-rootedness theorem, and inaccessible subsidiary results remain a novelty-audit limitation.

Who should care, and why

Why the problem matters

Real-rootedness links polynomial coefficients to strong structural inequalities. Ordinary coefficient powers look simple but need not preserve that structure for every real exponent. An exact endpoint identifies where the universal guarantee begins, and the equality case identifies its sharp boundary. The significance is a focused mathematical classification, not a claim of field-wide impact or an all-degree solution.

How to inspect or reproduce the recorded checks

Download the immutable evidence ZIP and its outer checksum. Verify the ZIP's SHA-256, extract it into a fresh directory, and follow README.md. The pinned Python dependencies are SymPy and mpmath. Run verify_package.py --root . --replay; the expected receipt is PASS for package integrity and scoped producer replay.

Read EVIDENCE_MAP.md before interpreting the output. The distinct checks of the analytical argument are in the manuscript and review reports, not hidden inside that status word. Rebuilding the PDF is optional for reading the proof and requires Pandoc and pdfLaTeX; the archived PDF has its own exact hash.

The most valuable next projects

1. Reconstruct the compactness and quartic boundary classification outside the producer workflow, and publish a scoped review or correction.
2. Complete the inaccessible-source and broader citation audit, including direct reconciliation with the originating AIM question.
3. Investigate a higher-degree critical-exponent theory. Degree five needs a threshold-relative scale argument and additional multiplicity cases; the present induction does not directly apply.
4. Formalize the analytical proof if the theorem survives specialist scrutiny.

The stretch assessment identifies the higher-degree programme as potentially substantial, but it supplies neither a higher-degree theorem nor a credible six-hour four-star closure estimate. It therefore does not enlarge this release's claim.

What is in the evidence package

The package contains the manuscript PDF, accessible Markdown and TeX source; the exact algebra checker, integrity verifier and negative controls; pinned requirements and Linux CI; structured claims and a machine entry point; source, novelty, licensing and provenance records; the supplied-review response; frozen internal editorial reports; a stretch assessment; and a complete file manifest. Outer ZIP checksums and fresh-extraction replay receipts are supplied as explicit sidecars. Original prose and records are CC0; original executable code is MIT. Cited third-party material and the externally supplied review are not relicensed or bundled.

@misc{sharpquartichadamardpowers2026,
  title        = {The sharp exponent for ordinary Hadamard powers of quartics with positive coefficients},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22346503},
  url          = {https://doi.org/10.5281/zenodo.22346503},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/sharp-quartic-hadamard-powers/}
}