Linear Total Differential Polynomials

• Linear total differential polynomials are defined as polynomials in which derivatives appear linearly using the radial differential operator across several complex variables.
• They enable explicit elimination and uniqueness results by linking algebraic differential elimination with Nevanlinna theory through determinantal and super-essentiality methods.
• These polynomials underpin value distribution and growth analyses, offering critical insights into Picard-type theorems and differential uniqueness in entire and meromorphic functions.

A linear total differential polynomial is a central object in the analysis of differential and value-distribution properties of meromorphic and entire functions, extending the classical theory of ordinary differential polynomials to higher dimensions and total derivatives. Formally, it is a polynomial whose monomials are linear combinations of all derivatives (in the sense of total or radial differentiation) of a function, with coefficients that are themselves small entire functions relative to the growth of the function in question. Linear total differential polynomials govern explicit elimination, uniqueness, and value-sharing phenomena and define bridges between algebraic differential elimination and Nevanlinna theory in several complex variables (Rueda, 2011, Ahamed et al., 18 Jan 2026).

1. Definitions and Structural Properties

A linear total differential polynomial in the sense of several complex variables is defined via the radial (total) differential operator

$D f(z) = \sum_{i=1}^n z_i \partial_{z_i} f(z)$

on the domain $\mathbb{C}^n$, with higher powers given recursively: $D^0 f = f$, $D^j f = D(D^{j-1} f)$ for $j \ge 1$. This yields

$$D^j f = \left( \sum_{i=1}^n z_i\partial_{z_i} \right)^j f = \sum_{|\alpha|=j} \frac{j!}{\alpha_1! \dots \alpha_n!} z^\alpha \partial^\alpha f$$

where $\alpha = (\alpha_1, \dots, \alpha_n)$ is a multi-index.

A linear total differential polynomial of order $k$ with "small" coefficient functions is then specified as

$$\mathcal{L}_k[D] f(z) = \sum_{j=1}^k a_j(z)\, D^j f(z)$$

where each $a_j(z)$ is entire with $T(r,a_j) = o(T(r,f))$ as $r \to \infty$, and $a_k \not\equiv 0$. This definition generalizes the classical linear ODE polynomial in one variable and incorporates all partial derivatives of total order $j$ for each term $D^j f$ (Ahamed et al., 18 Jan 2026).

By contrast, a general total differential polynomial (not necessarily linear) is an arbitrary polynomial in the infinite family of all derivatives of the variables, with coefficients in a suitable differential domain. The linearity distinguishes the class by ensuring all monomials have total degree at most one in the derivatives (Rueda, 2011).

2. Elimination Theory and Determinantal Formulas

In the context of systems of linear nonhomogeneous ordinary differential polynomials, elimination of differential indeterminates exploits determinantal formulas derived from coefficient matrices formed by appropriate derivatives ("slices") of the initial system. Given a system $\mathcal{P} = \{ f_1, \ldots, f_n \}$ in $n - 1$ indeterminates $U = \{ u_1, \ldots, u_{n-1} \}$, each

$f_i(U) = a_i + \sum_{j=1}^{n-1} L_{i,j}(u_j)$

with $a_i$ in a differential domain $D$, and $L_{i,j} \in D[\partial]$.

The eliminant, called the linear sparse differential resultant $\operatorname{dfres}(\mathcal{P})$, is given by

$$\operatorname{dfres}(\mathcal{P}) = \det \mathcal{M}(\mathcal{P})$$

where $\mathcal{M}(\mathcal{P})$ is an explicitly constructed square matrix whose rows are indexed by derivatives of the original equations and whose columns correspond to a chosen set of algebraic indeterminates and constants. The specific structure of $\mathcal{M}(\mathcal{P})$ ensures representability of the elimination ideal as an explicit determinant, provided certain "super-essentiality" conditions are satisfied (Rueda, 2011).

3. Super-Essentiality, Perturbation, and Nonvanishing Criteria

A system is called super-essential if, for each equation, there exists a permutation of the indeterminates ensuring that every associated differential operator is nonzero in a prescribed fashion, thus preventing zero columns in the coefficient matrix and ensuring the feasibility of explicit elimination. Formally, for each $i$, a permutation $\sigma_i$ of $\{1,\ldots,n-1\}$ is required so that $L_{j, \sigma_i(j)} \neq 0$ for $j \neq i$, and $L_{j, \sigma_i(j-1)} \neq 0$ for $j > i$. Under these conditions, the square matrix has full rank, and the resultant is generically nonzero. If super-essentiality fails, a linear perturbation is introduced to the system via auxiliary parameters to restore this property; the unperturbed resultant is obtained by extracting the coefficient of the zero-degree term in the auxiliary parameter expansion (Rueda, 2011).

4. Growth, Value Distribution, and Milloux Inequalities

For meromorphic (or entire) functions in $\mathbb{C}^n$, growth and value-distribution properties of their linear total differential polynomials are governed by extended Milloux-type inequalities. For transcendental meromorphic $f$,

$$T(r,f) \le N(r,0;f) + N(r,1;\mathcal{L}_k[D]f) - N(r,0;\mathcal{L}_{k+1}[D]f) + O(\log(r T(r,f)))$$

where $T(r,f)$ is the Nevanlinna characteristic, and $N(m,a;g)$ counts zeros or poles. This generalizes the one-variable classical inequality by incorporating the total/radial derivative structure.

Further, log-derivative estimates provide

$$m\bigl(r,\infty;\,D^j f/f\bigr) = O(\log(r T(r,f)))$$

and for the characteristic function,

$T(r,\mathcal{L}_k[D] f) \le (k+1+o(1))\, T(r, f)$

for transcendental meromorphic $f$ (Ahamed et al., 18 Jan 2026).

5. Uniqueness, Value-Sharing, and Picard-type Theorems

Critical applications of linear total differential polynomials appear in uniqueness theory and Picard-type statements. When two nonconstant meromorphic functions $f, g$ in $\mathbb{C}^n$ share $0$, $\infty$, and the value $1$ for their linear total differential polynomials $\mathcal L_k[D] f$, $\mathcal L_k[D] g$, and provided that the defect sum $2\delta(0, f) + (k+4)\Theta(\infty, f) > k+5$, the ratio

$$\frac{\mathcal{L}_k[D]f - 1}{\mathcal{L}_k[D]g - 1}$$

is necessarily a non-zero constant.

Additionally, a suite of Picard-type results holds: If $f$ is entire in $\mathbb{C}^n$ omitting a value $a$, and its linear total differential polynomial $\mathcal{L}_k[D]f$ omits a nonzero value $b$, then $f$ must be constant. This extends to polynomial-perturbed total differential polynomials and covers the omitting of single nonzero values by entire maps composed with polynomial expressions and the radial derivative (Ahamed et al., 18 Jan 2026).

6. Sparse Generic Systems and Relation to Sparse Differential Resultants

In the algebraic context, the sparse generic case involves monomial support sets $G_{i,j}$ and generic linear combinations. If the system is differentially essential, it admits a unique super-essential subsystem whose sparse linear differential resultant coincides, up to scalar multiple, with the differential resultant introduced by Li, Gao, and Yuan. This correspondence is exact under appropriate dimension counts and nonvanishing determinants, unifying determinantal elimination constructions with the established sparse resultant paradigm (Rueda, 2011).

7. Examples and Illustrative Computations

Explicit constructions are given for low-dimensional cases:

• For $n=2$, $U = \{u\}$, and equations involving combinations of $u$ and its derivatives, the matrix construction and resultant computation reproduce concrete linear relations in the elimination ideal, nontrivial for generic coefficients.
• For $n=3$, $U = \{u_1,u_2\}$, and equations with mixed partials, the methodology scales with the increase in variables and derivatives, systematically yielding a square matrix whose determinant is nonzero under genericity.

Further, in the function-theoretic context, examples show that if an entire $f$ (in one or several variables) and its linear total differential polynomial omit appropriate values, constantness is a forced conclusion. Classical special cases (e.g., the one-variable Yi–Yang theorem) are recovered by specializing to $n=1$ and traditional higher derivatives (Rueda, 2011, Ahamed et al., 18 Jan 2026).