The determinantal ideals of extended Hankel matrices

TL;DR

This paper investigates determinantal ideals of extended Hankel matrices using Gröbner bases, providing solutions to membership problems, primary decompositions, and proving linear resolutions and quadratic Gröbner bases for Rees algebras.

Contribution

It introduces new methods for analyzing extended Hankel matrix ideals, including solving membership problems and describing their algebraic properties.

Findings

Solved the membership problem for symbolic powers of the ideals.

Computed the primary decomposition of products of determinantal ideals.

Proved that these products have linear resolutions and their Rees algebras are defined by quadratic Gröbner bases.

Abstract

In this paper, we use the tools of Gr\"{o}bner bases and combinatorial secant varieties to study the determinantal ideals $I_t$ of the extended Hankel matrices. Denote by $c$-chain a sequence $a_1,\...,a_k$ with $a_i+c<a_{i+1}$ for all $i=1,\...,k-1$. Using the results of $c$-chain, we solve the membership problem for the symbolic powers $I_t^{(s)}$ and we compute the primary decomposition of the product $I_{t_1}\... I_{t_k}$ of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product $I_{t_1}\... I_{t_k}$ has a linear resolution and the multi-homogeneous Rees algebra $\Rees(I_{t_1},\...,I_{t_k})$ is defined by a Gr\"obner basis of quadrics.