Brüdern-Robert 丢番图系统中的强稀疏性
数论
2026-01-09 v1
摘要
设 k k k k ≥ 2 k\ge 2 k ≥ 2 ε > 0 \varepsilon>0 ε > 0 x 1 2 j − 1 + … + x k + 1 2 j − 1 = y 1 2 j − 1 + … + y k + 1 2 j − 1 ( 1 ≤ j ≤ k ) , x_1^{2j-1}+\ldots +x_{k+1}^{2j-1}=y_1^{2j-1}+\ldots +y_{k+1}^{2j-1}\quad (1\le j\le k), x 1 2 j − 1 + … + x k + 1 2 j − 1 = y 1 2 j − 1 + … + y k + 1 2 j − 1 ( 1 ≤ j ≤ k ) , 1 ≤ x i , y i ≤ P 1\le x_i,y_i\le P 1 ≤ x i , y i ≤ P ( 1 ≤ i ≤ k + 1 ) (1\le i\le k+1) ( 1 ≤ i ≤ k + 1 ) V k ∗ ( P ) V_k^*(P) V k ∗ ( P ) L k ∗ ( P ) L_k^*(P) L k ∗ ( P ) { x 1 , … , x k + 1 } = { y 1 , … , y k + 1 } \{x_1,\ldots ,x_{k+1}\}=\{y_1,\ldots ,y_{k+1}\} { x 1 , … , x k + 1 } = { y 1 , … , y k + 1 } L k ∗ ( P ) ∼ ( k + 1 ) ! P k + 1 L_k^*(P)\sim (k+1)!P^{k+1} L k ∗ ( P ) ∼ ( k + 1 )! P k + 1 V k ∗ ( P ) − L k ∗ ( P ) ≪ P 8 k + 9 − 1 + ε . V_k^*(P)-L_k^*(P)\ll P^{\sqrt{8k+9}-1+\varepsilon}. V k ∗ ( P ) − L k ∗ ( P ) ≪ P 8 k + 9 − 1 + ε .
引用
@article{arxiv.2601.05121,
title = {Strong paucity in the Br\"udern-Robert Diophantine system},
author = {Trevor D. Wooley},
journal= {arXiv preprint arXiv:2601.05121},
year = {2026}
}
备注
16 pages