Research Problems in Function Theory — Problem 3.25

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.

For $P\in\mathbb R[x_1,x_2,x_3]$, harmonicity means $\Delta P=0$, with $\Delta=\partial_{x_1}^2+\partial_{x_2}^2+\partial_{x_3}^2$. Divisibility by $Q=x_1^4+x_2^4+x_3^4$ means that $P=QR$ for some polynomial $R\in\mathbb R[x_1,x_2,x_3]$. Allowing complex coefficients gives the equivalent existence question by taking a nonzero real or imaginary part.

Statement recorded in the supplied source.

Is there a polynomial $R\not\equiv0$ such that \[\Delta\left((x_1^4+x_2^4+x_3^4)R\right)=0?\] Equivalently, can $Q$ divide a nonzero harmonic polynomial? [S2]

Short English statement

Can the sum of the fourth powers of three coordinates divide a nonzero harmonic polynomial?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303025 (AMR-022-3025). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 3.25, its complete Update 3.25, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.