论文标题
Gröbner基础用于增加序列
Gröbner Bases for Increasing Sequences
论文作者
论文摘要
令$ q，n \ geq 1 $为整数，$ [q] = \ {1，\ ldots，q \} $，$ \ mathbb f $是带有$ | \ mathbb f | \ geq q $的字段。增加序列$ i（n，q）= \ {（f_1，f_2，\ dots，f_n）\ in [q] $ j（n，q）$ affine space $ {\ mathbb f}^n $。我们描述了降低的Gröbner基础，标准单元和希尔伯特（Hilbert）的功能，这是多项式理想在$ j（n，q）$上消失的。 作为应用程序，我们给出了$ j（n，q）$的插值基础，以及增加Kakeya套件的大小，增加Nikodym集的下限，以及$ j（n，q）$的仿期超平面盖的大小。
Let $q,n \geq 1$ be integers, $[q]=\{1,\ldots, q\}$, and $\mathbb F$ be a field with $|\mathbb F|\geq q$. The set of increasing sequences $$ I(n,q)=\{(f_1,f_2, \dots, f_n) \in [q]^n:~ f_1\leq f_2\leq\cdots \leq f_n \} $$ can be mapped via an injective map $i: [q]\rightarrow \mathbb F $ into a subset $J(n,q)$ of the affine space ${\mathbb F}^n$. We describe reduced Gröbner bases, standard monomials and Hilbert function of the ideal of polynomials vanishing on $J(n,q)$. As applications we give an interpolation basis for $J(n,q)$, and lower bounds for the size of increasing Kakeya sets, increasing Nikodym sets, and for the size of affine hyperplane covers of $J(n,q)$.
