Nonlinear maps preserving the polynomial

Abstract: Let $\mathbb F$ be a field and $P \in \mathbb F [x_1,\ldots, x_n]$ be a homogeneous polynomial such that $|\mathbb F| &gt; °(P)$ and $$φ, ψ\colon \mathbb Fⁿ \to \mathbb F<sup>n$$ be two maps such that $$P(\mathbf{x} + λ\mathbf{y}) = P(φ(\mathbf{x}) + λψ(\mathbf{y}))$$ for all $λ\in \mathbb F$ and $\mathbf{x}, \mathbf{y} \in \mathbb F<sup>n.$ We provide the characterization of all such $φ$ and $ψ$ for all polynomials in the case if $\mathrm{char}(\mathbb F) = 0$ and for all polynomials satisfying certain condition in the case if $\mathrm{char}(\mathbb F) &gt; 0$. 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 $\mathcal L_{P} \subseteq {\mathbb F<sup>n}<sup>*$ 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 $φ, ψ$ 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 $k \ge 3$).

• The paper establishes that any pair of nonlinear maps preserving a homogeneous polynomial must coincide modulo its radical, enforcing a unique linear structure on the quotient space.
• It utilizes gradient-derived invariants and strict dimensional criteria to extend classical determinant and immanant preserver theorems to a broader class of polynomial invariants.
• Applications to matrix polynomial invariants, such as the Cullis determinant, demonstrate the theory’s practical impact on unifying transformational structures in algebra.

Nonlinear Maps Preserving Homogeneous Polynomial Structures

Introduction

The study of structure-preserving transformations, particularly those that maintain the values of polynomial invariants on algebraic or matrix spaces, is a central theme in linear and multilinear algebra. Classical results, such as Frobenius’s theorem, rigorously characterize linear maps that preserve the determinant on matrix algebras. Extensions of these results to broader classes of polynomial invariants and transformation types, in both linear and nonlinear settings, deepen our understanding of the underlying algebraic invariance. The paper "Nonlinear maps preserving the polynomial" (2604.23690) offers a comprehensive characterization of pairs of nonlinear maps $\phi, \psi : \mathbb{F}^n \to \mathbb{F}^n$ that satisfy, for a homogeneous polynomial $P$, the preservation relation

$$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$

for all $\lambda \in \mathbb{F}$ and all $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$, subject to field characteristic and degree constraints on $P$. The work generalizes the previous theory of determinant and immanant preservers to a broader class of polynomial invariants and explicates the implications for matrix polynomials, particularly the Cullis determinant.

Main Results and Characterization

The primary objective is to describe all pairs $(\phi, \psi)$ preserving a given homogeneous polynomial $P$ under the operation $\mathbf{x} + \lambda \mathbf{y}$. The preservation condition is sufficiently restrictive to enforce strong structural properties on $\phi$ and $P$0, even without imposing linearity or surjectivity. The analysis hinges on two critical algebraic constructs associated with $P$1:

• Radical $P$2: The set of directions along which $P$3 is invariant under translation; formally,

$P$4

• Space $P$5: The span of all functionals of the form $P$6, capturing the totality of first-order variations of $P$7; that is,

$P$8

The core theoretical results are:

1. Characteristic Zero Case: If $P$9 and $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$0 is homogeneous, then for any maps $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$1 satisfying the preservation condition, there exists a unique linear map $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$2 on the quotient space $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$3 such that both $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$4 and $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$5 reduce (modulo $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$6) to $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$7, and $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$8 preserves the induced polynomial $$P(\mathbf{x} + \lambda\mathbf{y}) = P(\phi(\mathbf{x}) + \lambda \psi (\mathbf{y}))$$9;
2. Positive Characteristic Case: If $\lambda \in \mathbb{F}$0 and $\lambda \in \mathbb{F}$1, and further the dimension criterion $\lambda \in \mathbb{F}$2 is satisfied, the same structural result holds.

This characterization strictly generalizes results for preservers of the determinant [Dolinar & Šemrl, Tan & Wang, Costara] as well as immanant preservers [Kuzma], unifying them under the broader framework of homogeneous polynomial invariants.

Structural Implications and Proof Strategy

The preservation relation imposes strong algebraic constraints: it forces $\lambda \in \mathbb{F}$3 and $\lambda \in \mathbb{F}$4 to coincide modulo $\lambda \in \mathbb{F}$5 and to reduce to a linear automorphism on the quotient, provided $\lambda \in \mathbb{F}$6 is nondegenerate in the appropriate sense. The dimension condition ensures that $\lambda \in \mathbb{F}$7 decomposes as a direct sum of $\lambda \in \mathbb{F}$8 and a subspace controlling the first-order behavior of $\lambda \in \mathbb{F}$9. The technical approach proceeds via:

• Identifying and analyzing the gradient-derived space $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$0, demonstrating its role in detecting degeneracies in $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$1's argument.
• Showing that any difference $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$2 must lie in $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$3, and both maps act linearly on the quotient.
• Employing classic field-theoretic results (e.g., if a nonzero polynomial of degree $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$4 over a field $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$5 with $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$6 vanishes everywhere, then it is the zero polynomial) to control the behavior of higher-degree preservers, especially in the positive characteristic and finite field regime.

These properties are shown to be tight: when the dimension or degree conditions fail, nontrivial nonlinear preservers can exist, but within the specified regime, only the described linear structure is possible.

Application to Matrix Polynomial Invariants: The Cullis Determinant

A concrete application involves the Cullis determinant $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$7, the classical alternating sum of maximal minors for $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$8 matrices. The characterization distinguishes between two regimes, depending on the parity of $\mathbf{x}, \mathbf{y} \in \mathbb{F}^n$9:

• $P$0 Even: The radical of $P$1 is zero (i.e., it is fully nondegenerate). All preservers are necessarily compositions of invertible linear changes of basis and column actions, matching the classical determinant case.
• $P$2 Odd: The radical is the space $P$3 of matrices with all rows equal. Preservers may include additional freedom to shift by elements in $P$4, but are otherwise constrained as in the even case.

Explicitly, the preservation relation

$P$5

forces, under suitable dimension and degree conditions, that $P$6 and $P$7 are of the form $P$8 modulo $P$9, where $(\phi, \psi)$0 satisfy explicit algebraic constraints determined by the structure constants of $(\phi, \psi)$1. This constitutes a significant advance over prior analyses, which were either limited to square matrices or restricted to the determinant or immanant.

Strong Results and Contrasting Claims

The paper's main numerical and algebraic findings are:

• Universality: Any pair of maps preserving the value of a homogeneous polynomial under the $(\phi, \psi)$2 operation is, modulo the radical, necessarily linear and uniquely determined by $(\phi, \psi)$3. This is a strong property characterizing the rigidity of polynomial invariance under nonlinear maps, conditional on the dimension and degree hypothese.
• Failure of Surjectivity: The requirement of surjectivity or linearity, crucial in previous theorems, is dropped. Even without the surjectivity assumption, rigidity is established via the algebraic properties of $(\phi, \psi)$4 and the field.
• Extension to Non-classical Determinants: By treating the Cullis determinant and its radical structure, the results handle rectangular matrices and incorporate previously unaddressed invariants.

No contradictory or unexpected behaviors are detected within the established hypotheses; the nonlinear preservation constraints act as a strong linearizing force under the specified algebraic conditions.

Implications and Future Directions

The results have potent implications in several domains:

• Operator Theory and Quantum Information: The characterization of structure-preserving nonlinear maps is directly relevant to questions regarding symmetries and conserved quantities in quantum systems, particularly where polynomial invariants underlie observable algebraic structures.
• Polynomial Invariant Theory: The criteria developed here provide immediate pathways for extending classification results to new polynomial invariants arising in commutative algebra, representation theory, and algebraic combinatorics.
• Algorithmic Applications: Given the ability to detect and classify preservers for general $(\phi, \psi)$5, these results may inform computational invariant theory and algorithms for recognizing isomorphisms or similarity transformations preserving polynomial relations.

The theoretical apparatus—especially the use of the gradient field span $(\phi, \psi)$6—opens programs for classifying more general classes of preservers, including those associated with systems of polynomials or with invariants on higher tensor spaces.

Prospective Research Problems

Several directions are highlighted for future investigation:

• Necessity of Hypotheses: The sharpness of the degree and dimension conditions is conjectured but not proven in full generality. Constructing counterexamples where these fail remains an open challenge.
• Polynomials with Exotic Radical Structures: More detailed classification of possible radicals and their interactions with $(\phi, \psi)$7 may yield finer distinctions, particularly in positive characteristic settings.
• Broader Classes of Invariants: Extensions to non-homogeneous or multihomogeneous polynomials, systems of invariants, or non-polynomial functions preserving similar relations are natural generalizations.

Conclusion

This work develops a unifying algebraic framework for describing all pairs of nonlinear maps preserving a given homogeneous polynomial under the addition-plus-scalar-multiplication action, contingent on field size and polynomial degree. The results subsume and extend a series of classical preserver theorems, providing both theoretical depth and concrete applications. The rigidity imposed by the preservation constraint—especially in characteristic zero or sufficiently large finite fields—forces linearity modulo the radical, yielding an explicit and constructive classification. By applying these results to non-classical matrix determinants, such as the Cullis determinant, the paper demonstrates both the power and generality of the approach, forming a basis for further innovation in invariant theory and its applications in mathematics and physics.