Gröbner bases for increasing sequences (Q6574394)

Let \(q,n\geqslant1\), \([q]:=\{1,\ldots,q\}\), \(\mathbb{F}\) be a field with an injection \(i:[q]\to\mathbb{F}\), \(I(n,q):=\{(f_1,\ldots,f_n)\in[q]^n:f_1\leqslant\cdots\leqslant f_n\}\), \(J(n,q)\) be the image of \(I(n,q)\) by the map \([q]^n\to\mathbb{F}^n\) induced by \(i\), and \(\mathbf{I}(J(n,q))\subseteq\mathbb{F}[x_1,\ldots,x_n]\) be the ideal of polynomials vanishing on \(J(n,q)\). The paper describes the reduced Gröbner bases, standard monomials, and Hilbert function of \(\mathbf{I}(J(n,q))\). As applications, it also gives the following results:\N\begin{itemize}\N\item[1.] A unique existence result of the polynomial interpolation on \(J(n,q)\) of degree \(q-1\).\N\item[2.] A lower bound on the number of hyperplanes covering \(J(n,q)-S\), where \(S\) is a subset of \(J(n,q)\) such that \(|S|\leqslant n\).\N\item[3.] A lower bound on the size of increasing Kakeya sets and increasing Nikodym sets. \(K\subseteq\mathbb{F}_q^n\) is called an increasing Kakeya set, if for each \(\mathbf{v}\in J(n,q)-\{\mathbf0\}\) there exists an \(\mathbf a\in\mathbb{F}_q^n\) such that \(\{\mathbf{a}+t\mathbf{v}:t\in\mathbb{F}_q\}\subseteq K\). This result strengthens that of \textit{Z. Dvir} [J. Am. Math. Soc. 22, No. 4, 1093--1097 (2009; Zbl 1202.52021)].\N\end{itemize}