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Tridimensional character sums with polynomial arguments and applications
Abstract
Let p be a large prime and χ a non-trivial Dirichlet character modulo p. We study the character sum ∑a ∼ A ∑b ∼ B ∑c ∼ C α(a,b) β(c)χ(f(a) + bc), where z ∼ Z means Z≤ z < 2Z, f ∈ ℤ[X] is of small degree k, and α=(α(a,b))a∼ A,b∼ B and β=(β(c))c∼ C are two complex coefficients. We prove non-trivial upper bounds for this sum in either of the two cases: (1) β≡1, k=2,3,4,5 and A,B,C>p1/8+ε, (2) β general, k=2,3 and A,B,C>p1/6+ε, where ε>0 is fixed. This work was originally motivated by an intermediate result of Ganguly and Rajan (2023) on counting 2×2 matrices over 𝔽p with irreducible characteristic polynomials, where the entries are in short segments. The new bounds here allow us to count such matrices in much shorter segments.
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BibTeX
@misc{fouvry2026tridimensional,
title = {{Tridimensional character sums with polynomial arguments and applications}},
author = {Étienne Fouvry and Igor E. Shparlinski and Ping Xi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01524},
eprint = {2610.01524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01524},
}
RIS
TY - UNPB
TI - Tridimensional character sums with polynomial arguments and applications
AU - Étienne Fouvry
AU - Igor E. Shparlinski
AU - Ping Xi
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01524
UR - https://arxiv.org/abs/2610.01524
AB - Let p be a large prime and χ a non-trivial Dirichlet character modulo p. We study the character sum ∑_a ∼ A ∑_b ∼ B ∑_c ∼ C α(a,b) β(c)χ(f(a) + bc), where z ∼ Z means Z≤ z < 2Z, f ∈ ℤ[X] is of small degree k, and α=(α(a,b))_a∼ A,b∼ B and β=(β(c))_c∼ C are two complex coefficients. We prove non-trivial upper bounds for this sum in either of the two cases: (1) β≡1, k=2,3,4,5 and A,B,C>p^1/8+ε, (2) β general, k=2,3 and A,B,C>p^1/6+ε, where ε>0 is fixed. This work was originally motivated by an intermediate result of Ganguly and Rajan (2023) on counting 2×2 matrices over 𝔽_p with irreducible characteristic polynomials, where the entries are in short segments. The new bounds here allow us to count such matrices in much shorter segments.
ER -