Brück-type conjecture in several complex variables for first-order partial derivatives

Prove that if f is a non-constant entire function on C^m with hyper-order ρ1(f) not in N ∪ {∞} and a ∈ C, and if for all i in {1,…,m} the relation ∂_{z_i} f(z) − a = e^{α(z)} (f(z) − a) holds with α an entire function on C^m, then α(z) is constant and f(z) = (c1/A) e^{A(z1+⋯+zm)} + a − a/A for non-zero constants A = e^{c} and c1.

Background

Motivated by the one-variable Brück conjecture and its exponential-ratio reformulation, the paper proposes an extension to several complex variables: if each first-order partial derivative of f shares a value with f in the CM sense through the exponential relation, then α should be constant and f should have a specific exponential form in the sum of coordinates.

The authors provide supporting results establishing the conjecture under additional assumptions: Theorem 3.1 proves the case a = 0 under the hyper-order restriction, and Theorem 3.2 proves the general a ≠ 0 case under a small-zero-count condition for each partial derivative. They also give examples showing the conjecture fails when the hyper-order is an integer or infinite, and when e{α(z)} is replaced by an entire function with zeros, indicating the necessity of the stated conditions.

References

Now motivated by Conjecture B, we suggest to extend Conjecture B into several complex variables as follows: Let $f$ be a non-constant entire function in $\mathbb{C}m$ such that $\rho_1(f)\not\in\mathbb{N}\cup{\infty}$ and $a\in\mathbb{C}$. If $\partial_{z_i}(f(z))-a=e{\alpha(z)}(f(z)-a)$, for all $i\in\mathbb{Z}[1,m]$, where $\alpha(z)$ is an entire function in $\mathbb{C}m$ and $a$ is a finite complex number, then $\alpha(z)$ reduces to a constant, $c$ say and $f(z)=\frac{c_1}{A}e{A(z_1+ \cdots+z_m)}+a-\frac{a}{A}$, where $A=ec$ and $c_1$ are non-zero constant.

— Bruck conjecture for solutions of first-order partial differential equations in Cm  (2509.02576 - Majumder et al., 25 Aug 2025) in Conjecture (Section 1, Introduction)