关于三角向量置换多项式幺半群、其单位群及其诱导置换群的结构
交换代数
2024-08-09 v2
摘要
设 n > 1 n>1 n > 1 R R R 1 ≠ 0 1\ne 0 1 = 0 R [ x 1 , … , x n ] n R[x_1,\ldots,x_n]^n R [ x 1 , … , x n ] n ( f 1 , … , f n ) (f_1,\ldots,f_n) ( f 1 , … , f n ) n n n f 1 , … , f n ∈ R [ x 1 , … , x n ] f_1,\ldots,f_n\in R[x_1,\ldots,x_n] f 1 , … , f n ∈ R [ x 1 , … , x n ] n n n g ⃗ ∘ f ⃗ = ( g 1 ( f 1 , … , f n ) , … , g n ( f 1 , … , f n ) ) , 其中 f ⃗ = ( f 1 , … , f n ) , g ⃗ = ( g 1 , … , g n ) \vec{g}\circ \vec{f}=(g_1(f_1, \ldots ,f_n), \ldots ,g_n (f_1, \ldots ,f_n)), \text{ 其中 }\vec{f}=(f_1, \ldots ,f_n), \vec{g}=(g_1, \ldots ,g_n) g ∘ f = ( g 1 ( f 1 , … , f n ) , … , g n ( f 1 , … , f n )) , 其中 f = ( f 1 , … , f n ) , g = ( g 1 , … , g n ) R [ x 1 , … , x n ] n R[x_1,\ldots,x_n]^n R [ x 1 , … , x n ] n f ⃗ = ( f 0 , f 1 + x 2 g 1 , … , f n − 1 + x n g n − 1 ) \vec{f}=(f_0,f_1 +x_2g_1,\ldots, f_{n-1} +x_n g_{n-1}) f = ( f 0 , f 1 + x 2 g 1 , … , f n − 1 + x n g n − 1 ) f 0 ∈ R [ x 1 ] f_0\in R[x_1] f 0 ∈ R [ x 1 ] R R R f i , g i ∈ R [ x 1 , … , x i ] f_i ,g_i\in R[x_1,\ldots,x_i] f i , g i ∈ R [ x 1 , … , x i ] g i g_i g i R i R^i R i R R R i = 1 , … , n − 1 i=1,\ldots, n-1 i = 1 , … , n − 1 R n R^n R n M T n \mathcal{MT}_n M T n f ⃗ \vec{f} f M T n \mathcal{MT}_n M T n f 0 f_0 f 0 R [ x 1 ] R[x_1] R [ x 1 ] R R R g i g_i g i R [ x 1 , … , x i ] R[x_1,\ldots,x_i] R [ x 1 , … , x i ] i = 1 , … , n − 1 i=1,\ldots, n-1 i = 1 , … , n − 1 R R R M T n \mathcal{MT}_n M T n R n R^n R n M T n \mathcal{MT}_n M T n n n n R R R
引用
@article{arxiv.2312.12099,
title = {On the structures of a monoid of triangular vector-permutation polynomials, its group of units and its induced group of permutations},
author = {Amr Ali Abdulkader Al-Maktry},
journal= {arXiv preprint arXiv:2312.12099},
year = {2024}
}