Gr\"obner Bases for Increasing Sequences

TL;DR

This paper studies the algebraic structure of increasing sequences by describing their vanishing ideals using Gr"obner bases, and applies these results to combinatorial problems like Kakeya sets and hyperplane covers.

Contribution

It provides explicit descriptions of Gr"obner bases, standard monomials, and Hilbert functions for the ideal of increasing sequences, with applications to combinatorial geometry.

Findings

Explicit Gr"obner bases for the ideal of increasing sequences.

Interpolation bases for the set of increasing sequences.

Lower bounds for combinatorial set sizes such as Kakeya and Nikodym sets.

Abstract

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 Gr\"obner 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 $J(n,q)$.