Least zero of pairs of additive cubic equations

2025/10/11 by Xiao, Yixiu, Li, Hongze
#11D45 #11P55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.10001

Abstract

An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms \begincases F = c₁x₁³ + c₂x₂³ + ⋯ + c_nx_n³ = 0,
G = d₁x₁³ + d₂x₂³ + ⋯ + d_nx_n³ = 0, \endcases under the "M-good" condition for n ≥ 16, where c₁, …, c_n and d₁, …, d_n are integers. Additionally, a range is derived for the probability that randomly selected simultaneous equations satisfy the M-good condition.

Citations

• On the number of points on a complete intersection over a finite field

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