A Nonmonotone Real-Rootedness Set for
Symmetric Imaginary Shifts
AI Research Artifact

Vasily Stodolsky Affiliation: Independent Researcher Affiliation: ORCID 0009-0003-5329-5308

Version 0.1.3, August 13, 2026
Authorship and provenance update

Abstract

For a real polynomial $F$ and $\omega\geq 0$, let

$$A_{\omega}(z)=\frac{F(z+i\omega)+F(z-i\omega)}{2},\hskip 20.00003pt\Omega_{F}=\{\omega\geq 0:A_{\omega}\text{ has only real zeros}\}.$$

This artifact presents an explicit rational even polynomial of degree eight for which $6/25$ and $12/25$ belong to $\Omega_{F}$, while $3/10$ does not. Exact Sturm certificates give respectively eight, four, and eight distinct real zeros. Consequently $\Omega_{F}$ is neither an interval nor an up-set. All zeros of $F$ lie in the strip $|\operatorname{Im}z|\leq 11/25$, and the classical strip-contraction theorem gives the eventual tail $[11/25,\infty)\subset\Omega_{F}$. We also include a direct elementary proof of that tail and a standard-library exact verifier.

Authorship and accountability. The theorem statements and candidate proofs were materially generated and revised with AI systems. Vasily Stodolsky is the named author, approved this version, and accepts responsibility for all its contents. The end-matter AI Use and Provenance Statement records the concrete human and model contributions. No independent human peer review is claimed.

Version note. Version 0.1.3 replaces mental-state and submission-decision wording with an action-based authorship and accountability statement and aligns Zenodo and arXiv metadata. The theorem statement, proof arguments, polynomial, and exact certificates are unchanged from version 0.1.2.

Keywords. Real-rooted polynomials; hyperbolic polynomials; symmetric imaginary shifts; Sturm theorem; finite-difference operators; zero-preserving operators.

MSC 2020. 26C10, 30C15, 47B38.

Persistent record. The version history is maintained at 10.5281/zenodo.21728324.

1 Introduction

Let $D$ denote differentiation. For a polynomial $F$, the operator

$$\cos(\omega D)F(z)=\frac{F(z+i\omega)+F(z-i\omega)}{2}$$

is a two-term finite-difference operator with imaginary shifts. Classical results of de Bruijn localize the zeros of such combinations and give quantitative contraction of a horizontal zero strip [5]. Brändén and Chasse give a precise modern strip-preserver formulation: if all zeros of a real polynomial lie in $|\operatorname{Im}z|\leq\mu$, then the output strip has half-width

$$\sqrt{\max\{\mu^{2}-\omega^{2},0\}};$$

see [3, Theorem 1.6]. In particular, every $\omega\geq\mu$ gives a real-rooted output.

The universal theory of hyperbolicity and stability preservers is much broader; see Borcea and Brändén [2]. Related finite-difference classifications and large-step asymptotics appear in Katkova, Tyaglov, and Vishnyakova [6]. Adams and Cardon study a symmetric imaginary-shift sum when the input already has only real zeros [1], while Cardon studies universal complex zero strip decrease for differential operators, including cosine symbols [4]. Nuij’s deformation $p+sp^{\prime}$ concerns paths and topology inside spaces of hyperbolic polynomials [7]. The classical multiplier-sequence framework of Pólya and Schur concerns diagonal coefficient operators [8]; it does not determine the fixed-input set $\Omega_{F}$ considered here.

These results control whole input classes or the eventual strip tail. A different question is what the parameter set

$$\Omega_{F}=\{\omega\geq 0:\cos(\omega D)F\text{ is real-rooted}\}$$

can look like for one fixed polynomial whose initial zeros are not all real. The example below shows that the set need not be monotone and need not be an interval, even when the input polynomial is real and even, its zeros lie in a bounded strip, and an eventual real-rooted tail is present.

The claim is deliberately narrow. No assertion of historical priority or minimal degree is made. A bounded source audit found no exact anticipation or stronger containment of the displayed finite witness. The complete 1914 Pólya-Schur article was inspected as background. One older monograph remains available only through an authoritative bibliographic record and modern formulations. The accompanying audit summary records that limit.

2 The polynomial and the main result

Define

$$F(z)={} \left(\left(z-\frac{1}{50}\right)^{2}+\left(\frac{7}{25}\right)^{2}\right)\left(\left(z+\frac{1}{50}\right)^{2}+\left(\frac{7}{25}\right)^{2}\right)$$

$$\hskip 10.00002pt\times\left(\left(z-\frac{3}{5}\right)^{2}+\left(\frac{11}{25}\right)^{2}\right)\left(\left(z+\frac{3}{5}\right)^{2}+\left(\frac{11}{25}\right)^{2}\right).$$ (1)

For $\omega\geq 0$, set

$$A_{\omega}(z)=\frac{F(z+i\omega)+F(z-i\omega)}{2},\hskip 20.00003pt\Omega_{F}=\{\omega\geq 0:A_{\omega}\text{ has only real zeros}\}.$$ (2)

Theorem 2.1.

The polynomial $F$ in (1) is real and even, and every zero of $F$ lies in $|\operatorname{Im}z|\leq 11/25$. Moreover,

$$\frac{6}{25}\in\Omega_{F},\hskip 20.00003pt\frac{3}{10}\notin\Omega_{F},\hskip 20.00003pt\frac{12}{25}\in\Omega_{F}.$$

The respective numbers of distinct real zeros of $A_{\omega}$ are $8$, $4$, and $8$. In addition,

$$\left[\frac{11}{25},\infty\right)\subset\Omega_{F}.$$

Consequently $\Omega_{F}$ is neither an up-set nor an interval.

Proof of the symmetry and strip assertions.

The zeros of $F$ are visible directly from (1):

$$\frac{1}{50}\mathbin{\pm}\frac{7}{25}i,-\frac{1}{50}\mathbin{\pm}\frac{7}{25}i,\frac{3}{5}\mathbin{\pm}\frac{11}{25}i,-\frac{3}{5}\mathbin{\pm}\frac{11}{25}i.$$

This proves the symmetry and strip assertions. The two remaining parts of the theorem are proved separately below. ∎

3 Proof idea and verification route

This section is an explanatory map to the formal certificate below. It does not add a theorem, replace the exact proof, or make a claim about the complete shape of $\Omega_{F}$. Its purpose is to make clear why one finite polynomial, three rational parameters, and exact Sturm computations suffice for the stated separation result.

Motivation and mechanism

The point of the three parameters is their order:

$$\frac{6}{25}=0.24<\frac{3}{10}=0.30<\frac{12}{25}=0.48.$$

At the first and third values, $A_{\omega}$ has all eight zeros real, while at the middle value it has only four real zeros and hence four nonreal zeros. An interval containing both $0.24$ and $0.48$ must contain the intermediate point $0.30$. Likewise, an up-set containing $0.24$ must contain every larger parameter, including $0.30$. The in-out-in pattern therefore rules out both an interval and a monotone threshold of good parameters. It is the pattern in one continuously parametrized family, not the individual root counts in isolation, that provides the counterexample. In particular, real-rootedness at one parameter and throughout an eventual tail cannot be joined by a purely structural monotonicity argument; additional information about the family is required. This is not presented as a refutation of a named conjecture, but as a separation result showing that the stated structural assumptions do not imply monotonicity.

The mechanism is deliberately finite. The polynomial is real and even, with four conjugate pairs at two rational horizontal locations and two rational strip heights. Symmetric imaginary translation preserves evenness. Thus the degree-eight question for $A_{\omega}$ can be re-expressed as a degree-four question in $t=z^{2}$. This reduction does not approximate the roots: it records the symmetry already present in the displayed polynomial. Positive roots in $t$ correspond in pairs to real roots in $z$, while a failure to obtain four positive roots prevents all eight roots of $A_{\omega}$ from being real.

Role of the assumptions and logical route

Reality ensures that the shift average has real coefficients, so Sturm’s theorem can count its real roots exactly. Evenness supplies the substitution $t=z^{2}$, reducing the certificate to quartics without changing its conclusion. The visible factor form of $F$ also locates its zeros in a strip of half-width $11/25$. That strip information is used only for the eventual-tail statement. It is not a substitute for the three finite certificates, and it does not imply monotonicity at smaller parameters.

The formal route is directional. First, the factorization establishes the symmetry and strip assertions. Second, exact expansion gives one quartic $P_{\omega}$ at each selected rational parameter. Third, a Sturm chain determines the number of positive roots of each quartic from endpoint sign variations. Fourth, the substitution $t=z^{2}$ converts those counts to the real-root counts of the degree-eight averages. Finally, the ordered in-out-in pattern yields the two set-theoretic consequences. The later tail argument is independent supporting information: it explains why an eventual real-rooted regime can coexist with the finite nonmonotone pattern.

Sturm certificates and the exact verifier

The exact certificate is shorter after using evenness. We write $A_{\omega}(z)=P_{\omega}(z^{2})$, where $P_{\omega}$ is a real quartic. Sturm’s theorem counts four, two, and four positive roots of these quartics. Every positive root $t$ produces the two real roots $z=\mathbin{\pm}\sqrt{t}$, so the corresponding degree-eight polynomials have eight, four, and eight real zeros. The expanded rational coefficients only keep this sign computation exact; the sign-variation table is the actual root-count certificate. The separate tail argument shows that all $\omega\geq 11/25$ are good, so the eventual classical regime coexists with nonmonotone behavior below it. The certificate is exact throughout: coefficients, Euclidean remainders, endpoint signs, and variation counts are rational computations. The accompanying standard-library verifier recomputes these objects using exact fractions. It is a reproducibility check for the displayed certificate, not a replacement for mathematical reading and not a claim of human peer review.

What this section does not claim

The argument establishes the displayed membership and non-membership facts, the corresponding root counts, and the stated eventual tail. It does not classify every parameter in $\Omega_{F}$, prove that degree eight is minimal, identify a first historical example, or decide monotonicity for a special family with additional structure. No numerical root approximation is used as the certificate. The AI-assisted origin and model-audit history are disclosed separately; they are provenance information and are not presented as an independent human referee report.

4 Exact Sturm certificate

Proof of the membership claims.

Because $F$ is even, every $A_{\omega}$ is even. Write

$$A_{\omega}(z)=P_{\omega}(z^{2}),$$

where $P_{\omega}$ is a real quartic. Exact expansion gives

$$P_{6/25}(t)={} t^{4}-\frac{2237}{1250}t^{3}+\frac{161441}{250000}t^{2}-\frac{5715517}{97656250}t+\frac{173989}{976562500},$$

$$P_{3/10}(t)={} t^{4}-\frac{3371}{1250}t^{3}+\frac{1333057}{1250000}t^{2}-\frac{267506531}{1953125000}t+\frac{906531301}{9765625000000},$$

$$P_{12/25}(t)={} t^{4}-\frac{1657}{250}t^{3}+\frac{5734597}{1250000}t^{2}-\frac{194818069}{244140625}t+\frac{1243480589}{122070312500}.$$

For any one of these quartics, define its Sturm chain by

$$S_{0}=P_{\omega},\hskip 20.00003ptS_{1}=P_{\omega}^{\prime},\hskip 20.00003ptS_{j+1}=-\operatorname{rem}(S_{j-1},S_{j}).$$

Exact Euclidean division over $\mathbb{Q}$ gives degrees $4,3,2,1,0$ in all three cases. The endpoint signs and variation counts are as follows. No entry in a displayed sign row is zero.

$\omega$	$\operatorname{sgn}S(0^{+})$	$V(0^{+})$	$\operatorname{sgn}S(+\infty)$	$V(+\infty)$	positive roots of $P_{\omega}$
$6/25$	$+\ -\ +\ -\ +$	$4$	$+\ +\ +\ +\ +$	$0$	$4$
$3/10$	$+\ -\ +\ -\ -$	$3$	$+\ +\ +\ +\ -$	$1$	$2$
$12/25$	$+\ -\ +\ -\ +$	$4$	$+\ +\ +\ +\ +$	$0$	$4$

The last member of each chain is a nonzero constant, so each quartic is square-free. Also $P_{\omega}(0)>0$. Sturm’s theorem therefore gives exactly four, two, and four distinct positive roots of the three quartics. Under $t=z^{2}$, each positive root yields the two distinct real roots $z=\mathbin{\pm}\sqrt{t}$. Thus the three polynomials $A_{\omega}$ have respectively eight, four, and eight distinct real roots. Since each $A_{\omega}$ has degree eight, the middle polynomial is not real-rooted. This proves the three membership claims. ∎

The exact verifier included with the release reconstructs $F$ from its four quadratic factors, expands all three vertical averages over $\mathbb{Q}$, computes both the quartic chains above and the degree-eight Sturm chains, and checks square-freeness and the root counts. It uses only the Python standard library.

5 The classical real-rooted tail

The containment

$$[11/25,\infty)\subset\Omega_{F}$$

is a special case of the de Bruijn strip-contraction theorem, in the precise form of [3, Theorem 1.6]. We include an elementary proof for this finite polynomial.

Proof of the tail.

Fix $\omega\geq 11/25$ and let $z=x+iy$ with $y>0$. For one conjugate pair $\beta\mathbin{\pm}i\gamma$ of zeros of $F$, put

$$a=(x-\beta)^{2},\hskip 20.00003ptg(s)=\bigl(a+(s-\gamma)^{2}\bigr)\bigl(a+(s+\gamma)^{2}\bigr).$$

The quantity $g(s)$ is the squared modulus contributed by this pair at imaginary coordinate $s$. Set $p=y-\omega$ and $q=y+\omega$. Direct subtraction gives

$$g(q)-g(p) =8y\omega\bigl(y^{2}+\omega^{2}+a-\gamma^{2}\bigr)$$

$$>0.$$

Indeed, every pair in (1) satisfies $|\gamma|\leq 11/25\leq\omega$, and $y>0$. Multiplying the strict inequality over the four conjugate pairs yields

$$|F(z+i\omega)|>|F(z-i\omega)|.$$

The two summands in (2) therefore cannot cancel in the upper half-plane. Since $A_{\omega}$ is a real polynomial, conjugation excludes zeros in the lower half-plane as well. Hence every zero of $A_{\omega}$ is real. ∎

6 Consequences and scope

Recall that a subset $U\subset[0,\infty)$ is an up-set if $\omega\in U$ and $\sigma>\omega$ imply $\sigma\in U$. Here

$$\frac{6}{25}<\frac{3}{10}<\frac{12}{25}.$$

Because $6/25\in\Omega_{F}$ but $3/10\notin\Omega_{F}$, the set $\Omega_{F}$ is not an up-set. Because both endpoints $6/25$ and $12/25$ belong to $\Omega_{F}$ while the intermediate point $3/10$ does not, $\Omega_{F}$ is not an interval.

The example refutes only a generic structural implication: real and even input, conjugation-symmetric strip zeros, and an eventual real-rooted tail do not force monotonicity of the fixed-input real-rootedness set. The result does not characterize all of $\Omega_{F}$, prove a lower bound on the degree of any such example, or address a distinguished transcendental or arithmetic family. Additional structure in a special family could still impose a monotonicity theorem absent from the finite example.

7 Reproducibility, AI use, and provenance

The release package contains the LaTeX source, this PDF, the standard-library exact verifier, a recorded verifier output, an audit summary, a bounded prior-art summary, and a SHA-256 manifest. Running

python -B tools/verify_d05_sturm.py

reproduces the coefficient maps, Sturm signs, variation counts, real-root counts, and square-freeness checks.

The explicit witness and candidate proof arose in a GPT-5.5/xhigh headless quota-failover continuation. Claude Opus 5 Max later audited and reconciled the source record; this was not the initial discovery. Fresh GPT-5.5/xhigh sessions were then used for independent proof reconstruction, repair verification, and post-audit cross-checking. GPT-5.6-sol/max was used for the focused prior-art adjudication. The exact arithmetic was also recomputed in fresh sessions and by the released deterministic script. Model cross-checking does not establish human peer review or historical priority. The accompanying workflow file gives the detailed roles and limitations.

Vasily Stodolsky directed the research workflow, set its scope and release criteria, curated the mathematical and verification record, made the final decisions on claims and revisions, and approved this version. He accepts responsibility for all contents of the work. The AI systems are research tools and are not listed as authors. No independent human peer review is claimed. Vasily Stodolsky is identified by ORCID 0009-0003-5329-5308.

License. The manuscript and documentation are licensed under CC BY 4.0. Executable code in the release is licensed under the MIT License.