Waring's Problem Over Finite Fields

Definitions and mathematical statement

The record contains two materially different Waring problems, which must not be identified.

For the finite-field question described in the database background, let $q$ be a prime power, ${\mathbb{F}}_q$ the field with $q$ elements, and $d\geq1$ an integer. A precise quantity corresponding to that background is \[ g_q(d)=\min\{r\geq1:\ \forall a\in{\mathbb{F}}_q\ \exists x_1,\ldots,x_r\in{\mathbb{F}}_q, \quad a=x_1^d+\cdots+x_r^d\}, \] with $g_q(d)=\infty$ if no such integer $r$ exists. Repetitions and zero summands are allowed, so "exactly $r$" also permits representations with fewer summands by adding zeros. The background asks for the minimum number of summands, or bounds for it, for given $d$ and $q$. Neither the stored imperative sentence nor its background supplies a particular bound, asymptotic regime, or hypotheses guaranteeing finiteness. This is a formalization of the background's finite-field question only.

By contrast, the supplied literature triage identifies the intended Ben Green source as current Problem 67, concerning the polynomial ring ${\mathbb{F}}_p[t]$, not the finite field ${\mathbb{F}}_q$. It describes a quantity $G_{\mathrm{fin}}(k)$ based on representations of high-degree polynomials as \[ f=h_1^k+\cdots+h_s^k,\qquad f,h_1,\ldots,h_s\in{\mathbb{F}}_p[t], \] with controlled degrees of the $h_i$. Here $p$ is prime, $k$ is the exponent, and $s$ counts summands; this differs from the background's use of $k$ for the number of $d$th powers. The supplied summary does not state the exact degree bound, admissible inputs, quantifiers in $p$, characteristic restrictions, or the defining minimization for $G_{\mathrm{fin}}(k)$. Accordingly, this display records the type of function-field representation and is not a complete definition of that source quantity.

The triage dated 2026-08-17 reports the Liu--Wooley bound $G_{\mathrm{fin}}(k)\leq(1+o(1))k\log k$ in the source formulation and Sawin's asymptotic result with $s=O(k)$ in sufficiently large characteristic, leading Green to mark current Problem 67 solved. The exact asymptotic and characteristic thresholds are not supplied here. The same triage describes the recovered function-field problem as having a strong 2024--2026 solution but expressly warns that this does not resolve the distinct finite-field question in the database background. The registry labels the record open, while the triage labels this defective combined record partially solved. A definitive statement and status require deciding which mathematical problem the record is to preserve.

Short English statement

The database asks for bounds on the number of $d$th powers needed to represent every element of a finite field. Its cited Green problem instead concerns degree-controlled sums of powers in a polynomial ring over a finite field and is reported solved. These are distinct questions, and the record must be repaired before it has a single precise target.

Sources

• Ben Green, 100 Open Problems, current Problem 67, marked Solved according to the supplied check dated 2026-08-17: https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf. The supplied triage attributes the definition of $G_{\mathrm{fin}}(k)$, the Liu--Wooley bound, and the account of Sawin's result to this source. This citation is retained without opening or generating a PDF.
• Will Sawin, "The asymptotic in Waring's problem over function fields via a singular locus in the circle method," arXiv:2412.14053, version 3 (2026), as supplied: https://arxiv.org/abs/2412.14053.
• Local registry data/unsolvedmath/problems.json, permanent catalogue entry, exact source code catalogue entry, "Waring's Problem Over Finite Fields," in Ben Green's 100 Open Problems; statement, background, and literature triage dated 2026-08-17. The stored code is retained despite the triage's identification with current Green Problem 67. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.