PreprintCC BY 4.0arXiv
Subconvexity of Short k-Free Exponential Sums in 𝔽q[t]
Abstract
We extend recent work of the author over ℤ into the positive characteristic setting of 𝔽q[t]. In particular, for a polynomial F ∈ 𝔽q[t] of degree N, let Rk(α) denote the exponential sum over k-free polynomials f with deg(f−F)<K. For all s>0, we prove essentially tight upper and lower bounds for the s-th moment of Rk(α) whenever K > (1/2+ε)N, and in even shorter intervals when s>1+1/k. As an application of these results, we prove a lower bound of order qK/6 for the L1-mean of the Möbius-twisted exponential sum over 𝔽q[t] whenever K >(1/2+ε)N.
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BibTeX
@misc{doyle2026subconvexity,
title = {{Subconvexity of Short $k$-Free Exponential Sums in $\mathbbF_q[t]$}},
author = {Ben Doyle},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30942},
eprint = {2608.30942},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30942},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Subconvexity of Short k-Free Exponential Sums in 𝔽_q[t]
AU - Ben Doyle
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30942
UR - https://arxiv.org/abs/2608.30942
AB - We extend recent work of the author over ℤ into the positive characteristic setting of 𝔽_q[t]. In particular, for a polynomial F ∈ 𝔽_q[t] of degree N, let R_k(α) denote the exponential sum over k-free polynomials f with deg(f−F)<K. For all s>0, we prove essentially tight upper and lower bounds for the s-th moment of R_k(α) whenever K > (1/2+ε)N, and in even shorter intervals when s>1+1/k. As an application of these results, we prove a lower bound of order q^K/6 for the L^1-mean of the Möbius-twisted exponential sum over 𝔽_q[t] whenever K >(1/2+ε)N.
ER -