Research Problems in Function Theory — Problem 4.4

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 nonzero polynomial $P(z)=\sum_{j\ge0}a_jz^j\in\mathbb C[z]$, let \[\tau(P)=\#\{j:a_j\ne0\}\] be its number of nonzero terms. Cancellation in the coefficients of $P^2$ is allowed. A term-count threshold is a function $f:\mathbb N\to\mathbb N$.

Statement recorded in the supplied source.

Does there exist such an $f$ with \[\tau(P)\ge f(k)\quad\Longrightarrow\quad\tau(P^2)\ge k \] for every $k\ge1$ and every nonzero complex polynomial $P$? If it exists, determine bounds for the least possible thresholds. [S2]

Short English statement

Can a bound on the number of terms in a square force a bound on the number of terms in the original polynomial, uniformly over all coefficients and exponents?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2304004 (AMR-022-4004). 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 4.4, its complete Update 4.4, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.