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The asymptotic in Waring's problem over function fields beyond twice the degree

  • Matthew Hase-Liu
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Abstract

We prove the expected asymptotic in Waring's problem over 𝔽q[T], with a power-saving error, whenever n>2d, the characteristic is greater than (d−1)2, and q satisfies an explicit lower bound. This range is sharp in general for the expected asymptotic uniformly in the target polynomial. Our main new input is an aggregate minor arc estimate: we count functionals according to the codimension of their associated singular loci and use intersection theory to bound the degrees of the resulting parameter spaces. In particular, if n≥ (2+ε)d, the required lower bound on q is polynomial in d of degree 2+4/ε.

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Cite

BibTeXRIS
BibTeX
@misc{haseliu2026asymptotic,
  title = {{The asymptotic in Waring's problem over function fields beyond twice the degree}},
  author = {Matthew Hase-Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11888},
  eprint = {2609.11888},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11888},
}
RIS
TY  - UNPB
TI  - The asymptotic in Waring's problem over function fields beyond twice the degree
AU  - Matthew Hase-Liu
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11888
UR  - https://arxiv.org/abs/2609.11888
AB  - We prove the expected asymptotic in Waring's problem over 𝔽_q[T], with a power-saving error, whenever n>2d, the characteristic is greater than (d−1)^2, and q satisfies an explicit lower bound. This range is sharp in general for the expected asymptotic uniformly in the target polynomial. Our main new input is an aggregate minor arc estimate: we count functionals according to the codimension of their associated singular loci and use intersection theory to bound the degrees of the resulting parameter spaces. In particular, if n≥ (2+ε)d, the required lower bound on q is polynomial in d of degree 2+4/ε.
ER  -