This paper proves a clean counting formula for a version of Waring’s problem over the polynomial ring F_q[T]. The authors show that, when the number of summands n is greater than twice the exponent d, and when the finite field F_q is large enough (and its characteristic p exceeds (d−1)^2), the number of ways to write a target polynomial as a sum of n dth powers has the expected leading size and a power‑saving error. The authors also show this threshold n>2d is the correct borderline in general.

Waring’s problem classically asks how many dth powers are needed to represent integers. Here the objects are polynomials over F_q, and the authors fix the exponent d and count n‑tuples of polynomials of degree at most e whose dth powers add to a fixed target polynomial f. The asymptotic is taken as the degree bound e grows. The main result gives an explicit formula: the count equals a natural main term of size about q^{e(n−d)+n−1} (built from local density factors) plus an error term smaller by a factor about q^{−e}. In plain terms, as e→∞ the main term predicts the number of representations very accurately, provided the field and number of variables meet the stated conditions.

The technical heart of the paper is a new treatment of the so‑called minor arcs in the circle‑method decomposition. Building on an earlier approach of Will Sawin, the authors group the minor‑arc functionals by the codimension h of their associated singular loci. They combine an estimate for complete exponential sums that worsens with smaller codimension (roughly |S(α)| ≤ (d−1)^h q^{e+1−h/2}) with a geometric count of how many functionals have codimension up to h. Intersection theory is used to control the degrees of the parameter spaces that parametrize these functionals. The outcome is an “aggregate” minor‑arc bound: either the number of functionals of codimension ≤h is about q^{d h}, or it is bounded by A_d^e q^{d h+1} with an explicit constant A_d = (12/5) d · 100^d. Combining these counts with the exponential‑sum bounds gives the required decay of the minor arcs when q satisfies an explicit lower bound; in particular, if n ≥ (2+ε)d the needed lower bound on q grows only polynomially in d, with degree 2 + 4/ε.