Nonlinear maps preserving the polynomial

arxiv: 2604.23690 · v1 · submitted 2026-04-26 · 🧮 math.CO

Nonlinear maps preserving the polynomial

Andrey Yurkov This is my paper

Pith reviewed 2026-05-08 05:54 UTC · model grok-4.3

classification 🧮 math.CO

keywords nonlinear mapshomogeneous polynomialspolynomial preserversgradient fielddeterminant preserversimmanantCullis determinantfield characteristic

0 comments p. Extension

The pith

The only nonlinear maps satisfying the polynomial preservation identity are those built from linear maps preserving the associated gradient space L_P.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper determines all pairs of maps φ and ψ from a vector space to itself that leave a fixed homogeneous polynomial invariant under the specified affine combination. The characterization is complete in characteristic zero for any such polynomial over a large enough field, and holds under an extra condition in positive characteristic. It builds on the introduction of a canonical linear space associated to the polynomial, namely the span of its gradient vectors. The work extends long-standing results on linear preservers of matrix functions like the determinant to a broader nonlinear context and applies the general theory to rectangular matrix invariants.

Core claim

For a homogeneous polynomial P with |F| > deg(P), the maps φ and ψ satisfy the polynomial preservation identity for all scalars λ if and only if they can be expressed in terms of linear transformations that leave invariant the vector space L_P spanned by the range of the gradient of P, with the exact form of the maps depending on whether the underlying field has characteristic zero or positive.

What carries the argument

The vector space L_P, which is the linear span of all gradients of the homogeneous polynomial P and serves as the key invariant controlling the possible preserving maps.

Load-bearing premise

The field must have more elements than the degree of the polynomial, and the polynomial must be homogeneous.

What would settle it

A counterexample consisting of a homogeneous polynomial P over a large field of characteristic zero together with maps φ and ψ satisfying the identity but not arising from any linear maps that preserve L_P would disprove the claimed characterization.

read the original abstract

Let $\mathbb F$ be a field and $P \in \mathbb F [x_1,\ldots, x_n]$ be a homogeneous polynomial such that $|\mathbb F| > \deg(P)$ and $\phi, \psi\colon \mathbb F^n \to \mathbb F^n$ be two maps such that $P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$ for all $\lambda \in \mathbb F$ and $\mathbf{x}, \mathbf{y} \in \mathbb F^n.$ We provide the characterization of all such $\phi$ and $\psi$ for all polynomials in the case if $\mathrm{char}(\mathbb F) = 0$ and for all polynomials satisfying certain condition in the case if $\mathrm{char}(\mathbb F) > 0$. This characterization generalizes the existing results regarding the linear maps on matrices preserving the determinant, the immanant and other homogeneous polynomial functions of matrix entries. To obtain the main result of this paper, we introduce the vector space $\mathcal L_{P} \subseteq {\mathbb F^n}^*$ spanned by the range of the gradient field of $P \in \mathbb F[x_1,\ldots, x_n]$. Being a linear invariant associated with $P,$ this space has several remarkable properties and may also be used for studying the linear maps preserving $P$. In addition, we demonstrate how the main result could be applied to the particular polynomial matrix invariants. Namely, we provide an explicit description of corresponding pairs of nonlinear maps $\phi, \psi$ for the case where $P$ is equal to the Cullis' determinant of $n\times k$ rectangular matrix (with the assumption that $n \ge k + 2$ and $k \ge 3$).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper cleanly characterizes nonlinear maps preserving homogeneous polynomials through a gradient-based invariant space L_P, extending some linear preserver results to the nonlinear setting.

read the letter

The core contribution is a description of all maps φ and ψ satisfying P(x + λ y) = P(φ(x) + λ ψ(y)) for homogeneous P over a field with |F| larger than the degree. They define L_P as the span of the image of the gradient map of P, then use the polynomial identity in λ to extract that φ and ψ must be affine with respect to this space in characteristic zero, and under an extra condition on P in positive characteristic. This recovers the known linear cases for the determinant and immanants as special instances when the maps are required to be linear. The construction of L_P is the genuinely new piece; it turns the functional equation into a linear-algebra problem on the dual space and looks reusable for other preserver questions. The application to Cullis' determinant on rectangular matrices gives a concrete illustration with explicit forms for φ and ψ when n ≥ k + 2 and k ≥ 3. The argument appears direct: expand both sides in λ, compare coefficients, and use the size of the field to avoid vanishing issues. The positive-characteristic case carries an additional hypothesis on P that the manuscript states explicitly, though it is not needed in characteristic zero. That hypothesis is the only real limitation; without it the result would cover more ground, but the paper does not claim it does. The work is aimed at people already working on polynomial preserver problems or matrix invariants. A reader who knows the linear theory will see immediately how L_P organizes the nonlinear extension. It is worth sending to referees because the central theorem is new, the hypotheses are stated up front, and the proof strategy is standard but applied in a fresh way. Minor polishing on the positive-characteristic statement would help, but the paper is ready for review.

Referee Report

0 major / 3 minor

Summary. The manuscript characterizes all maps ϕ, ψ : F^n → F^n satisfying the functional equation P(x + λ y) = P(ϕ(x) + λ ψ(y)) for every λ ∈ F and all x, y ∈ F^n, where P is a homogeneous polynomial over a field F with |F| > deg(P). In characteristic zero the result holds for every such P; in positive characteristic it holds for those P satisfying an additional (explicitly stated) condition. The proof proceeds by constructing the linear invariant L_P spanned by the image of the gradient map of P, substituting into the functional equation, and extracting linearity/affinity constraints on ϕ and ψ from the resulting polynomial identity in λ. The characterization is then specialized to the Cullis determinant of n × k rectangular matrices under the hypotheses n ≥ k + 2 and k ≥ 3.

Significance. If the stated hypotheses are met, the result supplies a uniform description of (possibly nonlinear) preservers for a broad class of homogeneous polynomials and recovers the classical linear preserver theorems for the determinant and immanant as special cases. The auxiliary space L_P is a new, canonically associated linear invariant that may be useful for other preserver problems. The concrete application to the Cullis determinant demonstrates that the abstract machinery yields explicit, computable forms for ϕ and ψ.

minor comments (3)

Abstract: the phrase 'certain condition' on P in positive characteristic is left undefined; a single-sentence description of the condition (or a forward reference to its precise statement in §2) would make the scope of the main theorem immediately clear to readers.
§1 (Introduction): the claim that the result 'generalizes the existing results regarding the linear maps … preserving the determinant, the immanant …' would be strengthened by a brief sentence contrasting the new nonlinear forms with the classical linear ones.
Notation: the definition of L_P as the span of the range of the gradient field is introduced in the abstract and used throughout; a displayed equation or short paragraph immediately after its first appearance would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript, the accurate summary of its contents, and the positive assessment of its significance. We are pleased that the referee recommends minor revision and will gladly incorporate any editorial or minor clarifications that may be suggested.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines the auxiliary space ℒ_P explicitly as the span of the image of the gradient map of the given homogeneous polynomial P, then substitutes into the functional equation P(x + λy) = P(ϕ(x) + λ ψ(y)) and extracts the form of ϕ and ψ by comparing coefficients of the resulting polynomial identity in λ. This proceeds directly from the stated hypotheses (|F| > deg(P), homogeneity, and the extra condition on P when char(F) > 0) without fitting any parameters to data, without renaming a known result, and without load-bearing self-citations. The argument therefore remains independent of its own outputs and does not reduce the claimed characterization to a tautology or to prior fitted quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The result rests on standard field axioms and the definition of homogeneous polynomials; the only new object is the gradient span L_P, which has no independent existence proof outside the paper.

axioms (2)

standard math F is a field
Used throughout to define vector spaces and polynomial rings.
domain assumption P is homogeneous of degree d with |F| > d
Stated as the setup for the functional equation.

invented entities (1)

L_P no independent evidence
purpose: Vector space spanned by the range of the gradient field of P
New linear invariant introduced to classify the maps phi and psi.

pith-pipeline@v0.9.0 · 5631 in / 1434 out tokens · 59984 ms · 2026-05-08T05:54:15.030301+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 2 canonical work pages · 1 internal anchor

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Guterman and A

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work page arXiv 2025
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work page internal anchor Pith review Pith/arXiv arXiv 2025
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A. Guterman and A. Yurkov. Linear maps preserving the Cullis’ determi- nant. III.Spec. Matrices, 2025.(submitted for publication)

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Y. Nakagami and H. Yanai. On Cullis’ determinant for rectangular matri- ces.Linear Algebra Appl., 422(2):422–441, 2007

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W.C.Waterhouse. Invertibilityoflinearmapspreservingmatrixinvariants. Linear Multilinear Algebra, 13(2):105–113, 1983

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[1] [1]

Amiri, M

A. Amiri, M. Fathy, and M. Bayat. Generalization of some determinantal identities for non-square matrices based on Radic’s definition.TWMS J. Pure Appl. Math., 1(2):163–175, 2010

2010

[2] [2]

C. Costara. Nonlinear maps preserving the elementary symmetric func- tions.Ars Mathematica Contemporanea, 21(1):P1.09, 2021

2021

[3] [3]

C. E. Cullis.Matrices and Determinoids: Volume 1. Calcutta University Readership Lectures. Cambridge University Press, 1913

1913

[4] [4]

Dolinar and P

G. Dolinar and P. Šemrl. Determinant preserving maps on matrix algebras. Linear Algebra and its Applications, 348(1–3):189–192, 2002

2002

[5] [5]

F. G. Frobenius.Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, pages 944–1015. S. B. Deutsch. Akad. Wiss. Berlin, 1897

[6] [6]

Guterman and B

A. Guterman and B. Kuzma. Preserving zeros of a polynomial.Commu- nications in Algebra, 37(11):4038–4064, 2009. 32

2009

[7] [7]

Guterman and A

A. Guterman and A. Yurkov. Linear maps preserving the Cullis’ determi- nant. I, 2025.Available athttps://arxiv.org/abs/2512.13437

work page arXiv 2025

[8] [8]

Linear maps preserving the Cullis' determinant. II

A. Guterman and A. Yurkov. Linear maps preserving the Cullis’ determi- nant. II, 2025.Available athttps://arxiv.org/abs/2512.13445

work page internal anchor Pith review Pith/arXiv arXiv 2025

[9] [9]

Guterman and A

A. Guterman and A. Yurkov. Linear maps preserving the Cullis’ determi- nant. III.Spec. Matrices, 2025.(submitted for publication)

2025

[10] [10]

B. Kuzma. A note on immanant preservers.Journal of Mathematical Sciences, 155(6):872–876, 2008

2008

[11] [11]

Lang.Algebra

S. Lang.Algebra. Springer New York, 2002

2002

[12] [12]

Nakagami and H

Y. Nakagami and H. Yanai. On Cullis’ determinant for rectangular matri- ces.Linear Algebra Appl., 422(2):422–441, 2007

2007

[13] [13]

Pierce et al

S. Pierce et al. A survey of linear preserver problems.Linear Multilinear Algebra, 33(1-2), 1992

1992

[14] [14]

and Wang F

Tan V. and Wang F. On determinant preserver problems.Linear Algebra and its Applications, 369:311–317, 2003

2003

[15] [15]

Invertibilityoflinearmapspreservingmatrixinvariants

W.C.Waterhouse. Invertibilityoflinearmapspreservingmatrixinvariants. Linear Multilinear Algebra, 13(2):105–113, 1983

1983

[16] [16]

P. Šemrl. Commutativity preserving maps.Linear Algebra and its Appli- cations, 429(5):1051–1070, 2008. 33

2008