Polynomial corners in finite fields beyond the distinct-degree case
Abstract
We prove a quantitative polynomial Roth theorem for corners in 𝔽p2 for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer d, there are constants p0 and C (depending only on p) so that for every $ p>p_0$, if polynomials ϕ1,ϕ2∈ \mathbbFp [y] are of degree ≤ d vanishing at $0$ and are not linearly dependent, then every A⊂𝔽p2 with $ |A|\ge C p^{2-1/14} $ contains a nontrivial corner (x1,x2), (x1+ϕ1(y),x2), (x1,x2+ϕ2(y)) for some y∈𝔽p×. This improves the estimate p2−1/16 of Han--Lacey--Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han--Lacey--Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map F̃:W→𝔸3 from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin--Schreier sheaf and Katz--Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An ℓ2 matrix estimate adapted to such sets completes the resonant case.
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BibTeX
@misc{li2026polynomial,
title = {{Polynomial corners in finite fields beyond the distinct-degree case}},
author = {Ji Li and Chun-Yen Shen and Tuyen Trung Truong and Liangchuan Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13891},
eprint = {2609.13891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13891},
}
RIS
TY - UNPB
TI - Polynomial corners in finite fields beyond the distinct-degree case
AU - Ji Li
AU - Chun-Yen Shen
AU - Tuyen Trung Truong
AU - Liangchuan Wu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13891
UR - https://arxiv.org/abs/2609.13891
AB - We prove a quantitative polynomial Roth theorem for corners in 𝔽_p^2 for arbitrary pairs of linearly independent polynomials. More precisely, given a positive integer d, there are constants p_0 and C (depending only on p) so that for every $ p>p_0$, if polynomials ϕ_1,ϕ_2∈ \mathbbF_p [y] are of degree ≤ d vanishing at $0$ and are not linearly dependent, then every A⊂𝔽_p^2 with $ |A|\ge C p^{2-1/14} $ contains a nontrivial corner (x_1,x_2), (x_1+ϕ_1(y),x_2), (x_1,x_2+ϕ_2(y)) for some y∈𝔽_p^×. This improves the estimate p^2−1/16 of Han--Lacey--Yang and removes the distinct-degree restriction from their quantitative theorem. The main obstruction is the equal-degree resonant case, where the Jacobian argument of Han--Lacey--Yang degenerates. We adjoin the frequency-independent part of the phase to form an augmented map F̃:W→𝔸^3 from the correlation threefold. We prove that this map is generically finite on every top-dimensional geometric component and has no two-dimensional fibre. Using the associated Artin--Schreier sheaf and Katz--Laumon estimates for Fourier transform of perverse sheaves, we obtain square-root cancellation outside an algebraic exceptional set of dimension at most one and uniformly bounded degree. A separate curve-sum argument gives uniform control on the exceptional set. An ℓ^2 matrix estimate adapted to such sets completes the resonant case.
ER -