Let $q,n \geq 1$ be integers, $[q]=\{1,\ldots, q\}$, and $F$ be a field with $|F|\geq q$. The setof 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 F $ into a subset $J(n,q)$ of the affine space $F^n$. We describe reduced Gröbner bases, standard monomials and Hilbert function of the ideal of polynomialsvanishing 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 $J(n,q)$.
@article{10_37236_11437,
author = {G\'abor Heged\"us and Lajos R\'onyai},
title = {Gr\"obner bases for increasing sequences},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {2},
doi = {10.37236/11437},
zbl = {1542.13023},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11437/}
}
TY - JOUR
AU - Gábor Hegedüs
AU - Lajos Rónyai
TI - Gröbner bases for increasing sequences
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11437/
DO - 10.37236/11437
ID - 10_37236_11437
ER -
%0 Journal Article
%A Gábor Hegedüs
%A Lajos Rónyai
%T Gröbner bases for increasing sequences
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/11437/
%R 10.37236/11437
%F 10_37236_11437
Gábor Hegedüs; Lajos Rónyai. Gröbner bases for increasing sequences. The electronic journal of combinatorics, Tome 31 (2024) no. 2. doi: 10.37236/11437
