论文标题
二项式理想和一致性$ \ mathbb {n}^n $
Binomial ideals and congruences on $\mathbb{N}^n$
论文作者
论文摘要
a \ emph {一致性}上的$ \ mathbb {n}^n $是$ \ mathbb {n}^n $上的等价关系，与加法结构兼容。如果$ \ bbbk $是一个字段，而$ i $是$ \ bbbk [x_1，\ dots，x_n] $中的\ emph {binmial precional}（即多项式产生的理想，最多可以使用两个术语），则是$ i $ faceces $ i $ facte $ \ nate $ \ nath $ \ n. n} $ \ n}如果存在$ \ MathBf {X}^{\ MathBf {u}} $和$ \ Mathbf {x}^{x}^{x}^{x}^{\ MathBf {v} $属于$ i $ i $ i $ i $ i $ i $的非零系数，则与$ \ mathbf {v} $等效等效。尽管以这种方式出现了$ \ mathbb {n}^n $的每个一致性，但这不是一对一的信件，因为许多二项式理想可能会引起相同的一致性。然而，二项式理想及其相应的一致性之间的联系很强，人们可能会将一致性视为二项式理想的基本组合结构。在当前的文献中，分别开发了$ \ mathbb {n}^n $的二项式理想和一致性理论。该调查论文的目的是提供详细的并行说明，该博览会提供代数直觉，以进行一致性的组合分析。为了阐述本调查文件，我们主要关注[Kahle and Miller，代数理论8（6）：1297-1364，2014]，着眼于[Eisenbud和Sturmfels。 Duke Math J 84（1）：1-45，1996]和[Ojeda和PiedraSánchez，J。Symbolic Comput 30（4）：383-400，2000]。
A \emph{congruence} on $\mathbb{N}^n$ is an equivalence relation on $\mathbb{N}^n$ that is compatible with the additive structure. If $\Bbbk$ is a field, and $I$ is a \emph{binomial ideal} in $\Bbbk[X_1,\dots,X_n]$ (that is, an ideal generated by polynomials with at most two terms), then $I$ induces a congruence on $\mathbb{N}^n$ by declaring $\mathbf{u}$ and $\mathbf{v}$ to be equivalent if there is a linear combination with nonzero coefficients of $\mathbf{X}^{\mathbf{u}}$ and $\mathbf{X}^{\mathbf{v}}$ that belongs to $I$. While every congruence on $\mathbb{N}^n$ arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on $\mathbb{N}^n$ are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly [Kahle and Miller, Algebra Number Theory 8(6):1297-1364, 2014] with an eye on [Eisenbud and Sturmfels. Duke Math J 84(1):1-45, 1996] and [Ojeda and Piedra Sánchez, J. Symbolic Comput 30(4):383-400, 2000].
