Spectral Incidence Bounds over Finite Fields
Abstract
Let Fq be a finite field, let F,G∈Fq[T] be nonconstant, and let J∈Fq[T]. We study the parameterized two-parameter family Ca,b={(x,t)∈Fq2:t=J(x)+aF(x)+bG(x)+λab}, (a,b)∈Fq2, where λ∈Fq×, with incidences counted with parameter multiplicity. If d=°F, e=°G, and charac(Fq)>d+e, we prove the uniform spectral estimate |I(cP,Ω)−|cP||Ω|/q| ≪d,eq5/8√(|cP||Ω|) for arbitrary cP,Ω⊆Fq2. The proof combines two additive Fourier transforms with a fourth-moment estimate for Kloosterman matrices. If 1≤ d,e≤2 and charac(Fq)>3, midpoint--difference coordinates and finite Weyl quantization sharpen the error to the Vinh scale O(q1/2√(|cP||Ω|)). We further show that the latter phenomenon is intrinsic to the coefficient curve rather than to the degree of a chosen parametrization. There is a polynomial Lüroth factorization F=f∘ H, G=g∘ H, Fq(f,g)=Fq(T), unique up to an affine change of H. If Γ is the plane curve parametrized properly by (f,g) and κH=maxs∈Fq|H−1(s)|, then finite-fibre pullback multiplies the centered incidence norm by at most √(κH). Consequently, if degΓ≤2 and charac(Fq)>3, then |I(cP,Ω)−|cP||Ω|/q| ≪ √(κH) q1/2√(|cP||Ω|), even when F and G have arbitrarily large raw degree. The resulting expansion estimates for J(x)+yF(x)+zG(x)+λyz hold without lower bounds on the individual input sets. In the balanced regime, the q3 missing-value scale for this family is already covered by a theorem of Arala--Chow; the incidence theorems here additionally yield expansion consequences in strongly unbalanced regimes.
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BibTeX
@misc{zhi2026spectral,
title = {{Spectral Incidence Bounds over Finite Fields}},
author = {Yao Zhi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03819},
eprint = {2610.03819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03819},
}
RIS
TY - UNPB
TI - Spectral Incidence Bounds over Finite Fields
AU - Yao Zhi
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03819
UR - https://arxiv.org/abs/2610.03819
AB - Let Fq be a finite field, let F,G∈Fq[T] be nonconstant, and let J∈Fq[T]. We study the parameterized two-parameter family C_a,b={(x,t)∈Fq^2:t=J(x)+aF(x)+bG(x)+λab}, (a,b)∈Fq^2, where λ∈Fq^×, with incidences counted with parameter multiplicity. If d=°F, e=°G, and charac(Fq)>d+e, we prove the uniform spectral estimate |I(cP,Ω)−|cP||Ω|/q| ≪_d,eq^5/8√(|cP||Ω|) for arbitrary cP,Ω⊆Fq^2. The proof combines two additive Fourier transforms with a fourth-moment estimate for Kloosterman matrices. If 1≤ d,e≤2 and charac(Fq)>3, midpoint--difference coordinates and finite Weyl quantization sharpen the error to the Vinh scale O(q^1/2√(|cP||Ω|)). We further show that the latter phenomenon is intrinsic to the coefficient curve rather than to the degree of a chosen parametrization. There is a polynomial Lüroth factorization F=f∘ H, G=g∘ H, Fq(f,g)=Fq(T), unique up to an affine change of H. If Γ is the plane curve parametrized properly by (f,g) and κ_H=max_s∈Fq|H^−1(s)|, then finite-fibre pullback multiplies the centered incidence norm by at most √(κ_H). Consequently, if degΓ≤2 and charac(Fq)>3, then |I(cP,Ω)−|cP||Ω|/q| ≪ √(κ_H) q^1/2√(|cP||Ω|), even when F and G have arbitrarily large raw degree. The resulting expansion estimates for J(x)+yF(x)+zG(x)+λyz hold without lower bounds on the individual input sets. In the balanced regime, the q^3 missing-value scale for this family is already covered by a theorem of Arala--Chow; the incidence theorems here additionally yield expansion consequences in strongly unbalanced regimes.
ER -