speaker: Zhuan John Ye, University of North Carolina Wilmington, Professor

time: 2024.5.29，3:00PM   

place: Tencent Meeting: 713-258-142

Abstract:

We start from the Pythagorean Theorem (gou gu Theorem in China) to the Fermat Last Theorem, then, to our current research results on partial differential and partial difference equations (PDDEs) in the complex space of dimension $n$.

Assume that $n$ is a positive integer, $p_j (j = 1,2,··· ,6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on C^n . Let $L(f) =\sum_{ j=1}^n q_{tj} f_{z_{t_ j}} and $f(z) = f(z_1 +c_1,...,z_n +c_n)$, where $q_{tj} (j = 1,2,··· ,s ≤ n)$ are non-zero polynomials on C^n and $c\in C^n\{0}$. We show the structures of all entire solutions to the non-linear PDDE

$$(p_1L(f) + p_2 \overline{f} + p_5 f) ^2 + (p_3L(f) + p_4 \overline{f} + p_6 f)^2 = p, $$

which is called a Fermat-type partial differential-difference equation. Further, we find many sufficient conditions and/or necessary conditions for the existence, as well as the concrete representations, of entire solutions to the Fermat-type PDDE. We also demonstrate several examples on C^2 with non-constant coefficients to verify that all representations in our theorems exist and are accurate and that the entire solutions to the Fermat-type PDDEs could have finite or infinite growth order. Many previous publications are corollaries of our theorems.