Ideals of the form $I_{1}(XY)$

TL;DR

This paper computes Gr"obner bases for determinantal ideals formed by the product of two matrices with indeterminate entries, enabling the determination of their Betti numbers.

Contribution

It introduces a method to compute Gr"obner bases for ideals of the form I_1(XY) and uses this to find Betti numbers, advancing algebraic understanding of such ideals.

Findings

Computed Gr"obner bases for I_1(XY) ideals.

Determined Betti numbers for these determinantal ideals.

Provided structural insights into the algebraic properties of I_1(XY).

Abstract

In this paper we compute Gr\"obner bases for determinantal ideals of the form $I_{1}(XY)$, where $X$ and $Y$ are both matrices whose entries are indeterminates over a field $K$. We use the Gr\"obner basis structure to determine Betti numbers for such ideals.