Research Problems in Function Theory — Problem 4.29

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 a polynomial $R\in\mathbb C[z]$, write $Z(R)=\{z\in\mathbb C:R(z)=0\}$, ignoring multiplicities. Let $P,Q$ be monic complex polynomials with \[Z(P)=Z(Q),\qquad Z(P')=Z(Q'). \] For a constant monic polynomial the polynomial is $1$ and its zero derivative has zero set $\mathbb C$; the nonconstant case is the substantive one.

Statement recorded in the supplied source.

Must there exist positive integers $a,b$ such that $P(z)^a\equiv Q(z)^b$ as polynomials? The assumptions require equality of zero sets, not equality of multiplicities; the source's special cases of at most five distinct common zeros or all common zeros collinear do not replace the general question. [S2]

Short English statement

If two monic polynomials have exactly the same zeros and their derivatives also have exactly the same zeros, must positive powers of the polynomials coincide?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304029 (AMR-022-4029). 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 4.29, its complete Update 4.29, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.