Research Problems in Function Theory — Problem 2.26

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Write $\mathcal P,\mathcal E,\mathcal R,\mathcal M$ for, respectively, the complex polynomials, entire functions, rational functions, and meromorphic functions on $\mathbb C$. For an integer $N\ge1$ and a class $\mathcal C$, let $g_{\mathcal C}(N)$ be the least $k$ for which every $f\in\mathcal C$ can be written as $f=\sum_{\nu=1}^k f_\nu^N$ with all $f_\nu\in\mathcal C$.

Statement recorded in the supplied source.

Determine $k(N)=g_{\mathcal E}(N)$, or obtain sharp estimates for it. In the extension explicitly posed in Update 2.26, determine or estimate $g_{\mathcal C}(N)$ for each $\mathcal C\in\{\mathcal P,\mathcal E,\mathcal R,\mathcal M\}$ and $N\ge2$. The same update asks about the associated identity \[1=\sum_{\nu=1}^k f_\nu^N\] with the summands $f_\nu^N$ linearly independent over $\mathbb C$. In particular, do there exist nonconstant meromorphic functions $f,g,h$ with $f^n+g^n+h^n=1$ for $n=7$ or $n=8$? [S2] The original entire-function problem has $k(N)\le N$ and $k(N)=N$ for $N=1,2,3$. The 2018 update explicitly broadens the classes and identifies the cases $n=7,8$ in the three-term equation. [S2]

Short English statement

Find how many powers are needed to express arbitrary functions in four analytic classes, and settle the specified three-term meromorphic Fermat equations.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302026 (AMR-022-2026). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.26 and Update 2.26. https://arxiv.org/abs/1809.07200.