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A Geometrical Characterization of Proportionally Modular Affine Semigroups
Results in Mathematics ( IF 1.1 ) Pub Date : 2020-06-13 , DOI: 10.1007/s00025-020-01230-3
J. D. Díaz-Ramírez , J. I. García-García , A. Sánchez-R.-Navarro , A. Vigneron-Tenorio

A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality $$f_1x_1+\cdots +f_nx_n \bmod b \le g_1x_1+\cdots +g_nx_n$$ f 1 x 1 + ⋯ + f n x n mod b ≤ g 1 x 1 + ⋯ + g n x n , where $$g_1,\dots ,g_n$$ g 1 , ⋯ , g n , $$f_1,\ldots ,f_n\in \mathbb {Z}$$ f 1 , … , f n ∈ Z , and $$b\in \mathbb {N}$$ b ∈ N . In this work, a geometrical characterization of these semigroups is given. On the basis of this geometrical approach, some algorithms are provided to check if a semigroup S in $$\mathbb {N}^n$$ N n , with $$\mathbb {N}^n{\setminus } S$$ N n \ S a finite set, is a proportionally modular affine semigroup.

中文翻译:

比例模仿射半群的几何表征

比例模仿射半群是模丢番图不等式 $$f_1x_1+\cdots +f_nx_n \bmod b \le g_1x_1+\cdots +g_nx_n$$ f 1 x 1 + ⋯ + fnxn mod b ≤ g 1 x 的一组非负整数解1 + ⋯ + gnxn , 其中 $$g_1,\dots ,g_n$$g 1 , ⋯, gn , $$f_1,\ldots ,f_n\in \mathbb {Z}$$ f 1 , ... , fn ∈ Z ,和 $$b\in \mathbb {N}$$ b ∈ N 。在这项工作中,给出了这些半群的几何特征。在这种几何方法的基础上,提供了一些算法来检查 $$\mathbb {N}^n$$ N n 中的半群 S 是否具有 $$\mathbb {N}^n{\setminus } S$$ N n\S 一个有限集,是一个比例模仿射半群。
更新日期:2020-06-13
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